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Phosphorylation
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It has been suggested that Protein phosphorylation be merged into this article. (Discuss) Proposed since September 2017.
A phosphorylated serine residue
Phosphorylation allows coupling and addition of a phosphoryl group. This particular biochemistry probably represents the origins of catecholamine overload (PO3)− to a molecule. In biology, phosphorylation and its counterpart, dephosphorylation, are critical for many cellular processes. A large fraction of proteins (between 1/3 to 2/3 of the proteome in eukaryotes [1][2]) are temporarily phosphorylated, as are many sugars, lipids, and other molecules. Phosphorylation is especially important for protein function as this modification activates (or deactivates) almost half of the enzymes, thereby regulating their function.[3][4][5] Protein phosphorylation is considered the most abundant post-translational modification in eukaryotes.
The prominent role of protein phosphorylation in biochemistry is illustrated by the huge body of studies published on the subject (as of March 2015, the MEDLINE database returns over 240,000 articles, mostly on protein phosphorylation).
Contents
[hide]
1 Glucose
2 Protein phosphorylation
2.1 Function
2.2 Signaling networks
2.3 Types of phosphorylation
2.4 Phosphorylation of histones
2.5 Detection and characterization
3 ATP
4 See also
5 References
6 External links
Glucose[edit]
Phosphorylation of sugars is often the first stage of their catabolism. It allows cells to accumulate sugars because the phosphate group prevents the molecules from diffusing back across their transporter. Phosphorylation of glucose is a key reaction in sugar metabolism because many sugars are first converted to glucose before they are metabolized further.
The chemical equation for the conversion of D-glucose to D-glucose-6-phosphate in the first step of glycolysis is given by
D-glucose + ATP → D-glucose-6-phosphate + ADP
ΔG° = −16.7 kJ/mol (° indicates measurement at standard condition)
Researcher D. G. Walker of the University of Birmingham determined the presence of two specific enzymes in adult guinea pig liver, both of which catalyze the phosphorylation of glucose to glucose 6 phosphate. The two enzymes have been identified as a specific glucokinase (ATP-D-glucose 6-phosphotransferase) and non-specific hexokinase (ATP-D-hexose 6-phosphotransferase).
Hepatic cell is freely permeable to glucose, and the initial rate of phosphorylation of glucose is the rate-limiting step in glucose metabolism by the liver (ATP-D-glucose 6-phosphotransferase) and non-specific hexokinase (ATP-D-hexose 6-phosphotransferase).[6]
The role of glucose 6-phosphate in glycogen synthase: High blood glucose concentration causes an increase in intracellular levels of glucose 6 phosphate in liver, skeletal muscle and fat (adipose) tissue. (ATP-D-glucose 6-phosphotransferase) and non-specific hexokinase (ATP-D-hexose 6-phosphotransferase). In liver, synthesis of glycogen is directly correlated by blood glucose concentration and in skeletal muscle and adipocytes, glucose has a minor effect on glycogen synthase. High blood glucose releases insulin, stimulating the trans location of specific glucose transporters to the cell membrane.[6][7]
The liver’s crucial role in controlling blood sugar concentrations by breaking down glucose into carbon dioxide and glycogen is characterized by the negative delta G value, which indicates that this is a point of regulation with. The hexokinase enzyme has a low Km, indicating a high affinity for glucose, so this initial phosphorylation can proceed even when glucose levels at nanoscopic scale within the blood.
The phosphorylation of glucose can be enhanced by the binding of Fructose-6-phosphate, and lessened by the binding fructose-1-phosphate. Fructose consumed in the diet is converted to F1P in the liver. This negates the action of F6P on glucokinase,[8] which ultimately favors the forward reaction. The capacity of liver cells to phosphorylate fructose exceeds capacity of metabolize fructose-1-phosphate. Consuming excess fructose ultimately results in an imbalance in liver metabolism, which indirectly exhausts the liver cell’s supply of ATP.[9]
Allosteric activation by glucose 6 phosphate, which acts as an effector, stimulates glycogen synthase, and glucose 6 phosphate may inhibit the phosphorylation of glycogen synthase by cyclic AMP-stimulated protein kinase.[7]
Phosphorylation of glucose is imperative in processes within the body. For example, phosphorylating glucose is necessary for insulin-dependent mechanistic target of rapamycin pathway activity within the heart. This further suggests a link between intermediary metabolism and cardiac growth.[10]
Protein phosphorylation[edit]
Main article: Protein phosphorylation
Function[edit]
Reversible phosphorylation of proteins is an important regulatory mechanism that occurs in both prokaryotic and eukaryotic organisms.[11][12][13][14] It is estimated that 230,000, 156,000 and 40,000 phosphorylation sites exist in human, mouse and yeast, respectively.[2] Kinases phosphorylate proteins and phosphatases dephosphorylate proteins. Many enzymes and receptors are switched "on" or "off" by phosphorylation and dephosphorylation. Reversible phosphorylation results in a conformational change in the structure in many enzymes and receptors, causing them to become activated or deactivated. Phosphorylation usually occurs on serine, threonine, tyrosine and histidine residues in eukaryotic proteins. Histidine phosphorylation of eukaryotic proteins appears to be much more frequent than tyrosine phosphorylation.[15] In prokaryotic proteins phosphorylation occurs on the serine, threonine, tyrosine, histidine or arginine or lysine residues.[11][12][15][16] The addition of a phosphate (PO4) molecule to a polar R group of an amino acid residue can turn a hydrophobic portion of a protein into a polar and extremely hydrophilic portion of a molecule. In this way protein dynamics can induce a conformational change in the structure of the protein via long-range allostery with other hydrophobic and hydrophilic residues in the protein.
One such example of the regulatory role that phosphorylation plays is the p53 tumor suppressor protein. The p53 protein is heavily regulated[17] and contains more than 18 different phosphorylation sites. Activation of p53 can lead to cell cycle arrest, which can be reversed under some circumstances, or apoptotic cell death.[18] This activity occurs only in situations wherein the cell is damaged or physiology is disturbed in normal healthy individuals.
Upon the deactivating signal, the protein becomes dephosphorylated again and stops working.[citation needed] This is the mechanism in many forms of signal transduction, for example the way in which incoming light is processed in the light-sensitive cells of the retina.
Regulatory roles of phosphorylation include
Biological thermodynamics of energy-requiring reactions
Phosphorylation of Na+/K+-ATPase during the transport of sodium (Na+) and potassium (K+) ions across the cell membrane in osmoregulation to maintain homeostasis of the body's water content.
Mediates enzyme inhibition
Phosphorylation of the enzyme GSK-3 by AKT (Protein kinase B) as part of the insulin signaling pathway.[19]
Phosphorylation of src tyrosine kinase (pronounced "sarc") by C-terminal Src kinase (Csk) induces a conformational change in the enzyme, resulting in a fold in the structure, which masks its kinase domain, and is thus shut "off".[20]
Important for protein-protein interaction via "recognition domains."
Phosphorylation of the cytosolic components of NADPH oxidase, a large membrane-bound, multi-protein enzyme present in phagocytic cells, plays an important role in the regulation of protein-protein interactions in the enzyme.[21]
Important in protein degradation.
In the late 1990s, it was recognized that phosphorylation of some proteins causes them to be degraded by the ATP-dependent ubiquitin/proteasome pathway. These target proteins become substrates for particular E3 ubiquitin ligases only when they are phosphorylated.
Important in glycolysis.
Glycolysis is an essential process of glucose degrading into two molecules of pyruvate, through various steps, with the help of different enzymes. It occurs in ten steps and proves that phosphorylation is a much required and necessary step to attain the end products. Phosphorylation initiates the reaction in step 1 of the preparatory step[22] (first half of glycolysis), and initiates step 6 of payoff phase (second phase of glycolysis).[23]
Glucose, by nature, is a small molecule with the ability to diffuse in and out of the cell. By phosphorylating glucose (adding a negatively charged phosphate group), glucose is converted to glucose-6-phosphate and trapped within the cell as the cell membrane is negatively charged. This reaction occurs due to the enzyme hexokinase, an enzyme that helps phosphorylate many six-membered ring structures. Glucose-6-phosphate cannot travel through the cell membrane and is therefore, coerced to stay inside the cell. Phosphorylation takes place in step 3, where fructose-6-phosphate is converted to fructose-1,6-bisphosphate. This reaction is catalyzed by phosphofructokinase.
While phosphorylation is performed by ATPs during preparatory steps, phosphorylation during payoff phase is maintained by inorganic phosphate. Each molecule of glyceraldehyde-3-phosphate is phosphorylated to form 1,3-bisphosphoglycerate. This reaction is catalyzed by GAPDH (glyceraldehyde-3-phosphate dehydrogenase). The cascade effect of phosphorylation eventually causes instability and allows enzymes to open the carbon bonds in glucose.
Phosphorylation functions as an extremely vital component of glycolysis, for it helps in transport, control and efficiency.[24]
Signaling networks[edit]
Elucidating complex signaling pathway phosphorylation events can be difficult. In cellular signaling pathways, protein A phosphorylates protein B, and B phosphorylates C. However, in another signaling pathway, protein D phosphorylates A, or phosphorylates protein C. Global approaches such as phosphoproteomics, the study of phosphorylated proteins, which is a sub-branch of proteomics, combined with mass spectrometry-based proteomics, have been utilised to identify and quantify dynamic changes in phosphorylated proteins over time. These techniques are becoming increasingly important for the systematic analysis of complex phosphorylation networks.[25] They have been successfully used to identify dynamic changes in the phosphorylation status of more than 6000 sites after stimulation with epidermal growth factor.[26] Another approach for understanding Phosphorylation Network, is by measuring the genetic interactions between multiple phosphorylating proteins and their targets. This reveals interesting recurring patterns of interactions – network motifs.[27] Computational methods have been developed to model phosphorylation networks [28][29] and predict their responses under different perturbations.[30]
Types of phosphorylation[edit]
For more details on this topic, see Kinase.
Within a protein, phosphorylation can occur on several amino acids. Phosphorylation on serine is the most common, followed by threonine. Tyrosine phosphorylation is relatively rare but is at the origin of protein phosphorylation signaling pathways in most of the eukaryotes. Phosphorylation on amino acids, such as serine, threonine, and tyrosine results in the formation of a phosphoprotein, when the phosphate group of the phosphoprotein reacts with the -OH group of a Ser, Thr, or Tyr sidechain in an esterification reaction.[31] However, since tyrosine phosphorylated proteins are relatively easy to purify using antibodies, tyrosine phosphorylation sites are relatively well understood. Histidine and aspartate phosphorylation occurs in prokaryotes as part of two-component signaling and in some cases in eukaryotes in some signal transduction pathways.[32] In prokaryotes, archea, and some lower eukaryotes histidine's nitrogen act as a nucleophile and binds to a phosphate group.[33] Once histidine is phosphorylated the regulatory domain of the response regulator catalyzes the transfer of the phosphate to aspartate.
Phosphorylation of histones[edit]
Eukaryotic DNA is organized with histone proteins in specific complexes called chromatin. The chromatin structure functions and facilitates the packaging, organization and distribution of eukaryotic DNA. However, it has a negative impact on several fundamental biological processes such as transcription, replication and DNA repair by restricting the accessibility of certain enzymes and proteins. Post-translational modification of histones such as histone phosphorylation has been shown to modify the chromatin structure by changing protein:DNA or protein:protein interactions.[34] Histone post-translational modifications modify the chromatin structure. The most commonly associated histone phosphorylation occurs during cellular responses to DNA damage, when phosphorylated histone H2A separates large chromatin domains around the site of DNA breakage.[35] Researchers investigated whether modifications of histones directly impact RNA polymerase II directed transcription. Researchers choose proteins that are known to modify histones to test their effects on transcription, and found that the stress-induced kinase, MSK1, inhibits RNA synthesis. Inhibition of transcription by MSK1 was most sensitive when the template was in chromatin, since DNA templates not in chromatin were resistant to the effects of MSK1. It was shown that MSK1 phosphorylated histone H2A on serine 1, and mutation of serine 1 to alanine blocked the inhibition of transcription by MSK1. Thus results suggested that the acetylation of histones can stimulate transcription by suppressing an inhibitory phosphorylation by a kinase as MSK1.[36]
Detection and characterization[edit]
Antibodies can be used as powerful tool to detect whether a protein is phosphorylated at a particular site. Antibodies bind to and detect phosphorylation-induced conformational changes in the protein. Such antibodies are called phospho-specific antibodies; hundreds of such antibodies are now available. They are becoming critical reagents both for basic research and for clinical diagnosis.
Example of posttranslational modification detected on a 2D gel (spot boundaries delimited by analysis software, identification by mass spectrometry, P46462 is the protein ID in Expasy)
Posttranslational modification (PTM) isoforms are easily detected on 2D gels. Indeed, phosphorylation replaces neutral hydroxyl groups on serines, threonines, or tyrosines with negatively charged phosphates with pKs near 1.2 and 6.5. Thus, below pH 5.5, phosphates add a single negative charge; near pH 6.5, they add 1.5 negative charges; above pH 7.5, they add 2 negative charges. The relative amount of each isoform can also easily and rapidly be determined from staining intensity on 2D gels.
In some very specific cases, the detection of the phosphorylation as a shift in the protein's electrophoretic mobility is possible on simple 1-dimensional SDS-PAGE gels, as it's described for instance for a transcriptional coactivator by Kovacs et al.[37] Strong phosphorylation-related conformational changes (that persist in detergent-containing solutions) are thought to underlie this phenomenon. Most of the phosphorylation sites for which such a mobility shift has been described fall in the category of SP and TP sites (i.e. a proline residue follows the phosphorylated serine or threonine residue).
More recently large-scale mass spectrometry analyses have been used to determine sites of protein phosphorylation. Over the last 4 years, dozens of studies have been published, each identifying thousands of sites, many of which were previously undescribed.[38][39] Mass spectrometry is ideally suited for such analyses using HCD or ETD fragmentation, as the addition of phosphorylation results in an increase in the mass of the protein and the phosphorylated residue. Advanced, highly accurate mass spectrometers are needed for these studies, limiting the technology to labs with high-end mass spectrometers. However, the analysis of phosphorylated peptides by mass spectrometry is still not as straightforward as for “regular”, unmodified peptides. Recently EThcD has been developed combining electron-transfer and higher-energy collision dissociation. Compared to the usual fragmentation methods, EThcD scheme provides more informative MS/MS spectra for unambiguous phosphosite localization.[40]
A detailed characterization of the sites of phosphorylation is very difficult, and the quantitation of protein phosphorylation by mass spectrometry requires isotopic internal standard approaches.[41] A relative quantitation can be obtained with a variety of differential isotope labeling technologies.[42] There are also several quantitative protein phosphorylation methods, including fluorescence immunoassays, Microscale thermophoresis, FRET, TRF, fluorescence polarization, fluorescence-quenching, mobility shift, bead-based detection, and cell-based formats.[43][44]
ATP[edit]
ATP, the "high-energy" exchange medium in the cell, is synthesized in the mitochondrion by addition of a third phosphate group to ADP in a process referred to as oxidative phosphorylation. ATP is also synthesized by substrate-level phosphorylation during glycolysis. ATP is synthesized at the expense of solar energy by photophosphorylation in the chloroplasts of plant cells.
See also[edit]
Moiety conservation
Phosida
RegPhos
Types of phosphorylation
References[edit]
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External links[edit]
Functional analyses for site-specific phosphorylation of a target protein in cells (A Protocol)
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Allysine-Allysine-Allysine-Lysine
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text:
We're Going to Be Friends
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"We're Going to Be Friends"
Single by The White Stripes
from the album White Blood Cells
Released
Late 2002
Format
CD
Recorded
Early 2001
Genre
Folk rock, acoustic rock
Length
2:28
Label
V2 Records
Songwriter(s)
Jack White
Producer(s)
Jack White
The White Stripes singles chronology
"Dead Leaves and the Dirty Ground"
(2002) "We're Going to Be Friends"
(2002) "Red Death at 6:14"
(2002)
"Dead Leaves and the Dirty Ground"
(2002) "We're Going to Be Friends"
(2002) "Red Death at 6:14"
(2002)
"We're Going to Be Friends" is a promotional single by the American alternative rock band White Stripes from their album White Blood Cells. It was released in late 2002, and tells the story of meeting a new friend at the beginning of a school year. Through its lyrics, it is able to evoke the simplicity and nostalgia of childhood.
Contents
[hide]
1 Lyrics
2 Themes
3 Video
4 Reception
5 In popular culture
6 Track listing
6.1 CD promo
7 Children's book adaptation
8 References
9 External links
Lyrics[edit]
"Suzy Lee," who is mentioned in the song in "We're Going to Be Friends", makes recurring appearances in White Stripes' discography, including in their eponymous album, which includes the song "Suzy Lee", as well as on Get Behind Me Satan,[1] which is dedicated to Suzy Lee, "Wherever she may be..."
Themes[edit]
The song speaks of a girl and boy who become friends while engaging in activities in and out of school. AllMusic said the song "takes a nostalgic look back at the innocence of school days with a surprisingly sensitive vocal as [Jack] expertly paints impressions of days past with deft economy."[2]
Video[edit]
The video features Jack and Meg White on a couch in front of a house at night time. Jack is playing a guitar, while Meg is sleeping alongside him. Jack wakes Meg as the song ends. The cover of the single is a still from the video.
Reception[edit]
Tom Maginnis with AllMusic called the song a "sweet acoustic ballad,"[2] and NME called it "fey childhood-sweetheart folk."[3] The A. V. Club said it was "the album's most shocking track."[4]
In popular culture[edit]
Featured in the opening credits of the 2004 film Napoleon Dynamite, writer/director Jared Hess commented that, originally, they could not find the song they were looking for, but wanted to use "We're Going to Be Friends," so a copy of Napoleon Dynamite was sent to The White Stripes, who promptly approved the song's use in the film.[citation needed]
Singer-songwriter Jack Johnson recorded a cover of the song on his album Sing-A-Longs and Lullabies for the Film Curious George. In 2008 and 2009, it was used in a promo for PBS Kids.[citation needed] Bright Eyes covered the song with guest First Aid Kit for the charity album Cool for School: For the Benefit of the Lunchbox Fund.[5]
This song was also played at the end of the House entitled "Mirror Mirror", which aired October 30, 2007, although the song is not included in the show's soundtrack. The song is also featured in the Life in Pieces episode "Sexting Mall Lemonade Heartbreak". CBS included a song in commercials for its 2006 program The Class that resembled the tune of "We're Going to Be Friends."[6] When magazine Broadcasting & Cable reached out to Monotone Management (the management company of the band at the time), the company declined to comment, but indicated that the band was aware of the song and were not pleased.[6] In subsequent commercials, CBS used a different song.[6]
This song was featured in the 2017 film Wonder. The film used two versions of the song, the original version performed by the White Stripes and a cover version performed by Caroline Pennell.[7]
This is one of Conan O'Brien's favorite songs by The White Stripes and at his request, they performed it on the final episode of Late Night with Conan O'Brien on February 20, 2009.[citation needed]
Track listing[edit]
"We're Going to be Friends"
"We're Going to be Friends" by The White Stripes.
Problems playing this file? See media help.
All songs by The White Stripes unless otherwise noted
CD promo[edit]
"We're Going to Be Friends"
Children's book adaptation[edit]
We're Going to Be Friends
Author
Jack White
Illustrator
Elinor Blake
Country
United States
Genre
Children's literature
Publisher
Third Man Books
Publication date
November 7, 2017
ISBN
9780996401692
Website
https://thirdmanbooks.com/books/friends/
In May 2017, Third Man Records announced a children's book based on the single to be released on November 21, 2017, shortly after The White Stripes’ 20th anniversary.[8] The book became available on November 7, 2017 for pre-release events and limited distribution.[9] It was illustrated by Elinor Blake,[8] an illustrator an animator for shows such as The Ren and Stimpy Show and Pee-wee's Playhouse.[9] The book will come with a digital copy of the original song and a cover version performed by The Woodstation Elementary School Singers,[8] as well as a cover by April March.[9]
References[edit]
Jump up ^ SARO (July 20, 2009). "#7 Suzy Lee". everyjackwhitesong.blogspot.com. Blogspot.com. Retrieved August 20, 2016.
^ Jump up to: a b Maginnis, Tom. "Song Review by Tom Maginnis". AllMusic. Retrieved 22 May 2017.
Jump up ^ NME staff. "The White Stripes : White Blood Cells". NME. Retrieved 22 May 2017.
Jump up ^ Phipps, Keith (Jul 3, 2001). "The White Stripes: White Blood Cells". A.V. Club. Retrieved 22 May 2017.
Jump up ^ "Listen: Bright Eyes and First Aid Kit Cover the White Stripes to Benefit South Africa". Cover Me. 4 October 2011. Retrieved 13 November 2017.
^ Jump up to: a b c Topcik, Joel (August 14, 2006). "Connecticut Candidate's Cable Connection". Broadcasting & Cable. 136 (32): 5.
Jump up ^ http://www.imdb.com/title/tt2543472/soundtrack
^ Jump up to: a b c Hughes, William (May 19, 2017). "Jack White is turning "We're Going To Be Friends" into a children's book". A. V. Club. Retrieved 22 May 2017.
^ Jump up to: a b c "We're Going to be Friends". Third Man Books. Retrieved 15 November 2017.
External links[edit]
White Stripes.Net FAQ. Retrieved September 17.
rollingstone.com
[hide]
v
t
e
The White Stripes
Jack White
Meg White
Studio albums
The White Stripes (1999)
De Stijl (2000)
White Blood Cells (2001)
Elephant (2003)
Get Behind Me Satan (2005)
Icky Thump (2007)
Live albums
Under Great White Northern Lights
Live in Mississippi
Video albums
Under Blackpool Lights
Under Great White Northern Lights / Under Nova Scotian Lights
Singles
"Let's Shake Hands"
"Lafayette Blues"
"The Big Three Killed My Baby"
"Hand Springs"
"Hello Operator"
"Lord, Send Me an Angel"
"Party of Special Things to Do"
"Hotel Yorba"
"Fell in Love with a Girl"
"Dead Leaves and the Dirty Ground"
"We're Going to Be Friends"
"Red Death at 6:14"
"Walking with a Ghost"
"Candy Cane Children"
"Seven Nation Army"
"I Just Don't Know What to Do with Myself"
"The Hardest Button to Button"
"There's No Home for You Here"
"Jolene (Live Under Blackpool Lights)"
"Blue Orchid"
"My Doorbell"
"The Denial Twist"
"Icky Thump"
"Rag and Bone"
"You Don't Know What Love Is (You Just Do as You're Told)"
"Conquest"
Other songs
"300 M.P.H. Torrential Outpour Blues"
"Ball and Biscuit"
"Suzy Lee"
Related articles
Discography
Songs
Awards and nominations
The Raconteurs
The Dead Weather
The Upholsterers
The Go
Two-Star Tabernacle
Aluminium
Third Man Records
Book
Category
Retrieved from "https://en.wikipedia.org/w/index.php?title=We%27re_Going_to_Be_Friends&oldid=836405024"
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text:
Candy pumpkin
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Candy Pumpkin
Candy corn and candy pumpkins
Type
Confectionery
Place of origin
United States
Main ingredients
Corn syrup, honey, carnauba wax, sugar
Cookbook: Candy Pumpkin
Media: Candy Pumpkin
A candy pumpkin is a small, pumpkin-shaped, mellowcreme confection primarily made from corn syrup, honey, carnauba wax, and sugar. Traditionally colored with an orange base and topped with a green stem[1] to make candy pumpkins largely identifiable with Halloween,[2] a candy pumpkin is considered a mellow creme by confectioners since the candy has a marshmallow flavor.[3] Sometimes called candy corn's first cousin,[4] candy pumpkins are made through a starch casting process similar to that for candy corn.[3] Brach's candy pumpkin, known by the trademarked name "Mellowcreme Pumpkin," is the most popular candy pumpkin.[3][5][6] Brach's Confections is now owned by Farley's & Sathers Candy Company.[7]
Contents
1 History
2 Impact
3 See also
4 References
History[edit]
Candy pumpkins are made using a similar process to make candy corn. The candy corn process and product were created by Géorge Renninger of the Wunderlee Candy Co. in the 1880s[8] and became popular as a treat in the 1920s.[3][9] Candy pumpkins first were produced in mid 20th century using a process similar to that of candy corn. Corn syrup, food coloring, honey, and sugar are beaten and heated in large kettles to produce an ultra-sweet syrup.[9] This syrupy mix generically is called "mellowcreme" by confectioners, since the resulting candy has a mellow, creamy texture.[5][9] The mellowcreme slurry then was divided into two uneven amounts, with the large amount receiving orange food coloring and the smaller receiving green food coloring.[9] A mogul machine brings the two colored mixtures together into a mold made of cornstarch, and the assembly is sent to a separate drying room to dry for 24 to 36 hours.[9] Once dry, the candy is shaken violently to remove excess cornstarch and a final glaze is added to give the candy pumpkin a sheen.[9] Candy pumpkins, acorns and other shapes that are derived from the mellowcreme mixture are often sold with candy corn under the name "harvest mix."[9]
Impact[edit]
Candy pumpkins are popular in part because the mellowcreme gives them "an interesting texture."[9] As of 1988, most big confectionery companies, including Mars Inc., did not market special Halloween candies.[10] The one exception was Brach's Confections, which made candy pumpkins among other seasonal products.[10] Their "Mellowcreme Pumpkin" was made to look like an autumnal fruit; each pumpkin contained 25 calories and 5 grams sugar.[11] In 1992, Brach's Confections expected to sell more than 30 million pounds of mellowcreme candy during the fall season, which included its seasonal mellowcreme pumpkins.[12] You can also find the candy pumpkin in Brach's Autumn Mix.
By the late 1990s, competitors of Brach's realized that the market for the special Halloween candy pumpkin was expanding. For example, in 1997, candy pumpkins and other mellowcreme candies helped push annual spending on Halloween candy in the United States to an estimated $950 million a year.[9] In response, Mars, Inc. came out with Snickers Creme Pumpkin in 1998.[13] The milk chocolate-covered peanut and caramel candy was packaged in a 1.20 oz. size with a plastic wrapper featuring a jack-o-lantern on the package.[13] At the time, the Snickers Creme Pumpkin retailed for 50 U.S. cents.[13] Two years later, in 2000, Frankford Candy & Chocolate Company cross-licensed with ConAgra Foods to produce Peter Pan Peanut Butter Pumpkins.[14] Peter Pan Peanut Butter Pumpkins included a "rich and creamy" Peter Pan peanut butter center pressed into a detailed pumpkin mold.[14] At that time, the Peter Pan pumpkin candy was sold in 14 oz. bags.[14] Also in 2000, Zachary Confections expanded its product line to include candy pumpkins.[15]
In addition to helping characterize Halloween, candy pumpkins played a role in the current U.S. implementation of daylight saving time. Since the 1960s, candy makers had wanted to get the trick-or-treat period covered by Daylight Saving, reasoning that if children have an extra hour of daylight, they would collect more candy.[16] During the 1985 U.S Congressional hearings on Daylight Saving, the industry went so far as to put candy pumpkins on the seat of every senator, hoping to win a little favor.[16] On July 8, 1986, President Ronald Reagan signed the Federal Fire Prevention and Control Act of 1986 into law; it contained a daylight saving rider which continued daylight saving time until the early morning of last Sunday in October;[16] this did not include Halloween night. In 2005, daylight saving time was extended to the first Sunday in November—just long enough to include Halloween.[17]
See also[edit]
Wikimedia Commons has media related to Candy pumpkins.
Food portal
Candy corn
References[edit]
Jump up ^ Flanigan, Kathy (September 21, 2001). "Pieces of Autumn - Raking Up A Pile of Ideas for the Season". E Cue. Milwaukee Journal Sentinel. p. 16. Retrieved 29 October 2008.
Jump up ^ Sulon, Bill (August 13, 2000). "Area Retailers Start Early with Seasonal Candy Displays". The Patriot-News. Retrieved 29 October 2008.
^ Jump up to: a b c d Blachford, Stacey L. (20 October 2008). "Candy Corn." How Products are Made. Retrieved 29 October 2008.
Jump up ^ Wheaton, Ken (October 29, 2007). "Happy Halloween ... and watch out for the chickens!". Advertising Age. 78 (43): 57. Retrieved 29 October 2008. [dead link]
^ Jump up to: a b "Brach's, Nation's Top Candy-Corn Maker, Scares Up Halloween Fun; Kids of All Ages to Consume 2 Billion 'Kernels' in 2004, Brach's 100th Birthday". PR Newswire Europe. October 13, 2004. Retrieved 29 October 2008.
Jump up ^ Miller, Lynne (17 September 2007). "Halloween presents supermarkets with a challenge to boost their candy market share". Supermarket News. Retrieved 29 October 2008.
Jump up ^ "Farley's & Sathers acquires Brach's". Candy & Snack Business. 2007-11. Retrieved 2008-07-27. Check date values in: |publication-date= (help)[dead link]
Jump up ^ Saeger, Natalie (29 October 2007). "History of candy corn. With new colors and flavors, a treat for all seasons". Showcase. The Spectator. Archived from the original on 24 January 2009. Retrieved 18 October 2008.
^ Jump up to: a b c d e f g h i Knauss, Christina Lee (October 29, 1997). "Stripped Candy Corn Now Symbolic of Fall". Food and Nutrition. Myrtle Beach Sun News. p. D1. Retrieved 29 October 2008.
^ Jump up to: a b Serlin, Bruce (October 23, 1988). "What's New in Halloween Marketing; Innovations in Fangs, Broomsticks and Candy Corn". 3. The New York Times. p. 315. Retrieved 29 October 2008.
Jump up ^ Murphy, Candace (October 30, 2006). "Kids say what matters about Halloween candy". Health Bay Area Living. Tri-Valley Herald. Retrieved 29 October 2008. [dead link]
Jump up ^ Lollar, Michael (October 15, 1992). "The Trick to Halloween Treat Game". Appeal. The Commercial Appeal. p. C2.
^ Jump up to: a b c "Snickers Chocolate Candy - Pumpkin". Product Alert by Marketing Intelligence Service Ltd. 28 (21). November 9, 1998.
^ Jump up to: a b c "Peter Pan Peanut Butter Pumpkins". Product Alert by Marketing Intelligence Service Ltd. 30 (11). June 12, 2000.
Jump up ^ Pacyniak, Bernard (May 1, 2006). "Staying fit". Candy Industry by Stagnito Publishing. 171 (5): 20. Retrieved 29 October 2008.
^ Jump up to: a b c Norris, Michele (March 8, 2007). "The Reasoning Behind Changing Daylight-Saving". All Things Considered. Retrieved 29 October 2008. BLOCK: This may be kind of an urban legend, but I thought I had heard that one of the backers behind extending Daylight Saving Time into the beginning of November was the candy industry, and it all had to do with Halloween. Mr. DOWNING: This is no kind of legend. This is the truth. For 25 years, candy-makers have wanted to get trick-or-treat covered by Daylight Saving, figuring that if children have an extra hour of daylight, they'll collect more candy. In fact, they went so far during the 1985 hearings on Daylight Saving as to put candy pumpkins on the seat of every senator, hoping to win a little favor.
Jump up ^ Allen, Kent (November 5, 2007). "A Busy Time on the Shuttle; Sweet and Light for the Candy Lobby; Cool Home Prices Yield Hot Auction; A Masterpiece in the Muck and Mire". U.S. News & World Report. 143 (16): 26. Kids on the prowl for candy this Halloween will have a bit more daylight in which to do it. That's because of tinkering by Congress that extended daylight saving time. Of course, the candy lobby was strongly in favor. In 1985, candy makers gave out candy pumpkins to members of Congress, hoping to curry sweet favors. In 2005, they succeeded. This year DST is lasting eight months, one month longer than in past years and just long enough to include Halloween.
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Set (mathematics)
From Wikipedia, the free encyclopedia
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This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory.
A set of polygons in a Venn diagram
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.
Contents
[hide]
1 Definition
2 Describing sets
3 Membership
3.1 Subsets
3.2 Power sets
4 Cardinality
5 Special sets
6 Basic operations
6.1 Unions
6.2 Intersections
6.3 Complements
6.4 Cartesian product
7 Applications
8 Axiomatic set theory
9 Principle of inclusion and exclusion
10 De Morgan's laws
11 See also
12 Notes
13 References
14 External links
Definition[edit]
Passage with a translation of the original set definition of Georg Cantor
A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]
A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.
Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.[2]
Cantor's definition turned out to be inadequate; instead, the notion of a "set" is taken as a primitive notion in axiomatic set theory, and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.
Describing sets[edit]
Main article: Set notation
There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:
A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.
The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}.
One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D.
There are two important points to note about sets. First, in an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}. However, per extensionality, two definitions of sets which differ only in that one of the definitions lists set members multiple times, define, in fact, the same set. Hence, the set {11, 6, 6} is exactly identical to the set {11, 6}. The second important point is that the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple). We can illustrate these two important points with an example:
{6, 11} = {11, 6} = {11, 6, 6, 11} .
For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as
{1, 2, 3, ..., 1000},
where the ellipsis ("...") indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.
The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have the meaning "the set of all ...". So, E = {playing card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A more general form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers that are four less than perfect squares can be denoted
F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}.
In this notation, the colon (":") means "such that", and the description can be interpreted as "F is the set of all numbers of the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the vertical bar ("|") is used instead of the colon.
Membership[edit]
Main article: Element (mathematics)
If B is a set and x is one of the objects of B, this is denoted x ∈ B, and is read as "x belongs to B", or "x is an element of B". If y is not a member of B then this is written as y ∉ B, and is read as "y does not belong to B".
For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 12 ∈ F; but
9 ∉ F and green ∉ B.
Subsets[edit]
Main article: Subset
If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.
If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).
Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas others use them to mean the same as A ⊊ B (respectively B ⊋ A).
A is a subset of B
Example:
The set of all men is a proper subset of the set of all people.
{1, 3} ⊆ {1, 2, 3, 4}.
{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set and every set is a subset of itself:
∅ ⊆ A.
A ⊆ A.
An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
A = B if and only if A ⊆ B and B ⊆ A.
A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.
Power sets[edit]
Main article: Power set
The power set of a set S is the set of all subsets of S. Note that the power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).
The power set of a finite set with n elements has 2n elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8 elements.
The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)
Every partition of a set S is a subset of the powerset of S.
Cardinality[edit]
Main article: Cardinality
The cardinality | S | of a set S is "the number of members of S." For example, if B = {blue, white, red}, | B | = 3.
There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and is denoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
Special sets[edit]
There are some sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted {} or ∅. Another is the unit set {x}, which contains exactly one element, namely x.[2] Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include
P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
N or N {\displaystyle \mathbb {N} } , denoting the set of all natural numbers: N = {0, 1, 2, 3, . . .} (sometimes defined excluding 0).
Z or Z {\displaystyle \mathbb {Z} } , denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
R or R {\displaystyle \mathbb {R} } , denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers that cannot be rewritten as fractions, such as √2, as well as transcendental numbers such as π, e and numbers that cannot be defined).
C or ℂ, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.
H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.
Positive and negative sets are denoted by a superscript - or +. For example, ℚ+ represents the set of positive rational numbers.
Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related fields.
Basic operations[edit]
There are several fundamental operations for constructing new sets from given sets.
Unions[edit]
The union of A and B, denoted A ∪ B
Main article: Union (set theory)
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
Some basic properties of unions:
A ∪ B = B ∪ A.
A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ⊆ (A ∪ B).
A ∪ A = A.
A ∪ ∅ = A.
A ⊆ B if and only if A ∪ B = B.
Intersections[edit]
Main article: Intersection (set theory)
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.
{1, 2} ∩ {2, 3} = {2}.
Some basic properties of intersections:
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B if and only if A ∩ B = A.
Complements[edit]
The relative complement
of B in A
The complement of A in U
The symmetric difference of A and B
Main article: Complement (set theory)
Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.
Examples:
{1, 2} \ {1, 2} = ∅.
{1, 2, 3, 4} \ {1, 3} = {2, 4}.
If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.
Some basic properties of complements:
A \ B ≠ B \ A for A ≠ B.
A ∪ A′ = U.
A ∩ A′ = ∅.
(A′)′ = A.
A \ A = ∅.
U′ = ∅ and ∅′ = U.
A \ B = A ∩ B′.
An extension of the complement is the symmetric difference, defined for sets A, B as
A Δ B = ( A ∖ B ) ∪ ( B ∖ A ) . {\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}
For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.
Cartesian product[edit]
Main article: Cartesian product
A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.
Examples:
{1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
{1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
{a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.
Some basic properties of Cartesian products:
A × ∅ = ∅.
A × (B ∪ C) = (A × B) ∪ (A × C).
(A ∪ B) × C = (A × C) ∪ (B × C).
Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:
| A × B | = | B × A | = | A | × | B |.
Applications[edit]
Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.
One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs (x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of all squares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x2. The reason these two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y, y2) is a member of the set F.
Axiomatic set theory[edit]
Main article: Axiomatic set theory
Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:
Russell's paradox—It shows that the "set of all sets that do not contain themselves," i.e. the "set" { x : x is a set and x ∉ x } does not exist.
Cantor's paradox—It shows that "the set of all sets" cannot exist.
The reason is that the phrase well-defined is not very well defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.
For most purposes however, naive set theory is still useful.
Principle of inclusion and exclusion[edit]
Main article: Inclusion–exclusion principle
This principle can be used to find the cardinality of the union of sets.
| A 1 ∪ A 2 ∪ A 3 ∪ … ∪ A n | = ( | A 1 | + | A 2 | + | A 3 | + … | A n | ) − ( | A 1 ∩ A 2 | + | A 1 ∩ A 3 | + … | A n − 1 ∩ A n | ) + … + ( − 1 ) n − 1 ( | A 1 ∩ A 2 ∩ A 3 ∩ … ∩ A n | ) . {\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)-\\&{}\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)+\\&{}\ldots +\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}
De Morgan's laws[edit]
Augustus De Morgan stated two laws about sets.
If A and B are any two sets then,
(A ∪ B)′ = A′ ∩ B′
The complement of A union B equals the complement of A intersected with the complement of B.
(A ∩ B)′ = A′ ∪ B′
The complement of A intersected with B is equal to the complement of A union to the complement of B.
See also[edit]
Logic portal
Set notation
Mathematical object
Alternative set theory
Axiomatic set theory
Category of sets
Class (set theory)
Dense set
Family of sets
Fuzzy set
Internal set
Mereology
Multiset
Naive set theory
Principia Mathematica
Rough set
Russell's paradox
Sequence (mathematics)
Taxonomy
Tuple
Notes[edit]
Jump up ^ "Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens – welche Elemente der Menge genannt werden – zu einem Ganzen." [1]
^ Jump up to: a b Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.
References[edit]
Wikimedia Commons has media related to Sets.
Dauben, Joseph W., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard University Press (1979) ISBN 978-0-691-02447-9.
Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.
Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4.
Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4
External links[edit]
C2 Wiki – Examples of set operations using English operators.
Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy
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