The ide of a collection is among the most fundamental ideas in chathamtownfc.netematics.Essentially, a collection is simply a arsenal of objects. The field of chathamtownfc.netematicsthat research studies sets, called collection theory, was started by the Germanchathamtownfc.netematician GeorgCantor in the latter fifty percent of the 19th century. Now the principle of setspermeates nearly all of modern-day chathamtownfc.netematics; almost every various other chathamtownfc.netematicalconcept (including the seemingly fundamental concept of numbers!) has beendefined, directly or indirectly, in regards to sets.

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## Basic definitions and notation

A set is a arsenal of objects, thought about as a chathamtownfc.netematicalobject in its very own right. (A helpful an allegory for a collection is a cardboardbox—the box deserve to hold objects, and we can select to think around theobjects in the box individually, or to think about the box and its contentscollectively together a solitary object.) The objects in a collection are referred to as theelements (or members) the the set; the elements are saidto belong come the collection (or to it is in in the set), and the setis stated to save the elements. Typically the aspects of a set areother chathamtownfc.netematics objects, such as numbers, variables, or geometricpoints.

### Writing sets

A set is regularly written by listing its aspects between curlybraces. For example, a set containing the numbers 1,2,and3 would certainly be created as 1,2,3.

When the aspects of a collection follow an obvious pattern yet there room too manyof lock to perform explicitly, that is common to list the first couple of elements (toestablish the pattern) and also the last facet (to specify whereby the patternstops), v an ellipsis(…) in between to show that theelements in the middle were omitted. For instance, to create the set of positiveintegers native 1 to100, we deserve to write1,2,3,…,100. If no element is written afterthe ellipsis, the pattern is presume to continue forever; therefore the collection written1,2,3,… contains every one of the hopeful integers.Sometimes the aspects of a set go on forever in both“directions”—for instance, the collection of all integers (bothpositive and negative) deserve to be composed as…,−3,−2,−1,0,1,2,3,….

A collection can also be defined in natural language utilizing English phrases. Forexample, “the set of all optimistic integers” describes a particularset. That is necessary that the summary be precise, so the there is no doubtabout even if it is a certain object is or is not an aspect of the set. A commonexample of one imprecise summary is a expression such as “the numbersbetween 1 and10.” This phrase has several ambiguities. Space thenumbers 1 and10 themselves aspects of the set, or space they excluded? Areall real numbers in between 1 and10 included, or only the integers?Finally, the phrase should more than likely be clearer around the truth that thesenumbers room to be taken into consideration as a set, quite than individually. A more precisedescription of this set might be “the set of optimistic integers no greaterthan10” (if the number 1 and10 are in the set, and also onlywhole numbers are included) or “the set of real numbers between 1and10, exclusive” (if the set includes numbers with fractionalparts, however not the number 1 and10 themselves).

The symbol∈ is provided to average “is an aspect of,” justas the symbol= is offered to typical “equals.” for example, to saythat the number2 is an aspect of the collection 1,2,3, we canwrite 2∈1,2,3. To express the opposite,“is no an element of,” we put a slash through the prize andwrite∉. For example, we can write4∉1,2,3.

It is often advantageous to assign names come sets. This names space usually chosento be solitary letters, comparable to the usage of letter to represent numbers inalgebra. A really common convention is to use resources letters for the name ofsets and lower-case letters to represent the facets of sets. So, forinstance, if we assign the letterA to the set1,2,3 by composing A=1,2,3, wecan climate say that 2∈A. Us can likewise writea∈A, through which we average thatais a (possibly unknown) number that is an aspect of thesetA—in other words, the value ofa mustbe either 1,2, or3.

### Equality the sets

Two set are said to be same if every aspect of the an initial setis an element of the second set, and vice versa. Because that example, ifA=1,2,3 andB=2,3,1, climate the set AandB space equal. Every aspect ofA is additionally anelement ofB, and also every element ofB is likewise anelement ofA. We express collection equality v an equals sign, soin this situation we create A=B. As soon as two to adjust areequal, they are thought about to it is in the exact same set. That is important to mental thatequality in between sets is a different concept than equality between numbers.

One consequence of this definition of equality is the the stimulate in whichthe elements of a set are provided is irrelevant. We care only about whichobjects are elements of the set, no the order castle come in. So, for instance,the expression 1,2,3 and 2,3,1 both define thesame set.

Another repercussion is the the number of times an aspect is noted isirrelevant. The set 1,2,1,3,3,3 is equal to theset 1,2,3, since every facet of the an initial set is an elementof the second, and also vice versa. (It does not issue that the numbers 1and3 are detailed several times in the first set.)

isimportant. Together a collection is dubbed a succession or anordered list. An instance is the use of ordered bag ofthe kind (x,y) to represent points in atwo-dimensional plane; the point(2,5) is different from thepoint(5,2). Similarly, us sometimes need a collection of objects inwhich it is meaningful for an facet to appear an ext than once. Together acollection is referred to as amultiset. Whenwe refer to a set, however, it is come be interpreted that the order andrepetition the the elements is irrelevant.>

## Sets containing other sets

The aspects of a collection can be anything—even other sets. For example,suppose we have actually two sets, A=1,2,3 andB=2,3,4,5. Think that A asa box containing the numbers 1,2, and3, and Bas abox comprise the numbers 2,3, 4, and5. There is naught stoppingus from putting the boxA and also the boxBtogether within a bigger boxC. In the same way, we can make asetC containing the setA and thesetB as elements. We deserve to write the setC asA,B, or, if us like, we have the right to say

C=1,2,3,2,3,4,5.

A set containing various other sets is prefer a box containing various other boxes. In thiscase, the setC has actually two elements, which are the twosets A andB. These elements of thesetC room themselves sets; the setA has actually threeelements, and also the setB has 4 elements.

Therefore it is true that A∈C,because A is an element ofC. Similarly, it is truethat B∈C. The course, the is additionally truethat, say, 1∈A, since the number1 is anelement that the setA.

However, the is not true the 1∈C,because 1 is no an aspect of the setC. The just two elementsofC room the to adjust A andB, andneither of these two facets is the number1. (The setAcontains the number1, however Aitself is no thenumber1.)

Returning to the box metaphor, we must think around elements that a collection (abox) together the objects that are directly within the box. Thenumber1 is not straight inside the boxC; instead, that ishidden within the boxA. For this reason 1is no an element of thesetC, but 1is an facet of thesetA.

Likewise, even though the is true that 2∈A andalso true the 2∈B, the is not true that2∈C, since the number2 is no directlyinside the boxC.

Now let’s take into consideration the set

D=1,2,3,2,3,4,5,

which is the very same as the set 1,2,3,4,5 due to the fact that theorder and also repetition the the facets are irrelevant. The setDis different indigenous the setC—these sets room notequal. To see this, that is sufficient to discover a single element in one set that isnot an facet of the other set. Well, the number1, because that example, is anelement ofD but (as defined above) is no an elementofC; for this reason the sets are not equal. Reasoning of CandD together boxes, we check out why they room not the same: thesetC is a crate that includes two crate (which themselves happento contain some numbers), whereas the setD is a crate thatdirectly consists of five number (and no boxes).

As an additional example, consider the 2 sets E=1 andF=1. These sets are not equal.The setE has actually one element, which is the number1; thesetF also has one element, however the single elementofF is one more set, no a number. So, whereas thesetE is a box containing the number1, thesetF is a box containing a crate containing the number1.(In various other words, the setF is a boxcontainingE—do you watch this?) Both of this aredifferent indigenous the number1 itself. So that is true that1∈E, and in factE∈F (because the single elementofF is the setE), however it is not truethat 1∈F.

Sets containing various other sets are typical in some locations of chathamtownfc.netematics, however wewon’t watch them very often in this course. (chathamtownfc.netematicians who occupational inthese areas often speak to a set containing various other sets a “collection ofsets” or a “family that sets” to stop the azer phrase“set of sets.”)

## Cardinality

The cardinality that a collection is the variety of elements that the setcontains. (Sometimes the cardinality that a collection is simply referred to as the“size” the the set, since size is ashorter word; however “cardinality” is the technically correct term.)

It is common to write|A| to typical the cardinality of thesetA. Because that example, ifA=1,4,8, then|A|=3, due to the fact that A has actually three elements. This isthe exact same notation as is offered for the absolute value of a number, however contextmakes it clean what is meant: if the upright bars surround a set, they referto the cardinality of the set.

The cardinality of a collection might be infinite. Because that instance, the cardinality ofthe collection of optimistic integers, 1,2,3,…, isinfinite.

## The north set

We have been thinking of sets together boxes. What type of set corresponds come theidea that an north box? plainly it should be a set that consists of no facets atall. We contact such a set an empty set. Since any type of two empty setscontain specifically the same elements (specifically, no aspects at all), weconsider any type of two north sets to it is in equal, and also hence the same set. So we usuallyrefer to the empty set, since there is really just one.

There space two common ways to create the empty set. The very first way, as youmight expect, is. The other notation is a one or a zero through aslash v it, i beg your pardon looks like∅. Bothand∅ space symbols because that the north set.

Note that there is a difference in between ∅ and∅. First is the empty set, i m sorry is an north box. The 2nd is a boxcontaining an north box, for this reason the 2nd box is no empty—it has actually abox in it! Don’t write∅ when you average the north set,because ∅ refers to a collection containing the north set, i m sorry is no thesame together the empty set itself.

The cardinality that the empty set is0, which provides sense, because theempty set contains no elements. The statementx∈∅ is constantly false, no matter whatx is, since there is no such point as an aspect of the emptyset.

## Subsets

Above, we identified two sets A andB to be same ifevery element ofA is an facet ofB, and also viceversa. If we eliminate the “vice versa” indigenous this definition, we getthe an interpretation of a subset.

We say the a setA is a subset of asetB if every element ofA is also an elementofB. We use the symbol⊆ to typical “is a subsetof“; for example, A⊆B means“Ais a subset ofB.” We can alsoturn this price the other method and create the to adjust in the other order to getB⊇A; this method the very same asA⊆B. To create that Aisnot a subset ofB, we draw a slash through the subset symbol:A⊈B. Intuitively, a subsetofB is a “part” ofB.

For example, consider the sets C=1,2 andD=1,2,3,4. The setCis a subset ofD, due to the fact that every facet ofC isalso an aspect ofD. Therefore we can writeC⊆D. A given set has plenty of subsets; forexample, one more subset ofD is 1,3,4.

To remember i m sorry direction to write the symbol⊆, think around thecardinalities of the sets and also the inequality symbol≤. If Ais a subset ofB, then B should contain at least asmany elements as Adoes (simply due to the fact that B containsall that the elements ofA), so the cardinalityofB should be better than or equal to the cardinalityofA. In other words,

A⊆Bimplies that|A|≤|B|.

Note the the symbols over “point” in the same direction.

There is a really important difference between the signs ∈and⊆. The symbol∈ is offered to indicate an elementof a set, conversely, the symbol⊆ is offered to show a subset.For instance, think about the set

D=1,2,3,4.

The set2,4 is a subset ofD, because everyelement of2,4 is additionally an facet ofD, so that iscorrect to compose 2,4⊆D. However theset2,4 is not an aspect of the setD,because the four aspects of the setD space all numbers(Ddoes not have any type of sets together elements), so that is untrue towrite 2,4∈D. Top top the various other hand, thenumber2 is an facet ofD, so it is exactly to write2∈D; however the number2 is not a subsetofD (because the number2 is a number, no a set), therefore itis not correct to create 2⊆D. If we want to to express tothe subset ofD containing just the number2, we shouldwrite2, i beg your pardon is the set containing2 (note that the set2is various from the number2). The set2 is a subsetofD, thatis, 2⊆D; but itis not an facet ofD, due to the fact that D walk nothave any sets together elements, so that is incorrect to write2∈D.

When A is a subset ofB, thesetB is occasionally dubbed a supersetofA. We can additionally say“BcontainsA together a subset,” butgreat care should be taken with words “contains,” because, asnoted in the ahead paragraph, over there is a large difference in between saying“BcontainsA as an element”(meaning A∈B) and also saying“BcontainsA as a subset” (meaningA⊆B). I will shot to usage the wordcontains just to refer to elements, no subsets.

A careful reading of the definition of subset shows that every collection is asubset of itself. Because that example, using the setD native above, itis obviously true that “every aspect ofD is additionally anelement ofD,” therefore by an interpretation Dis asubset ofD. We regularly want come exclude this case, for this reason we define aproper subset the a setB to it is in a subsetofB that is not Bitself. We writeA⊂B to median that Ais aproper subset ofB. notto it is in a subsetofB. Indigenous the definition, this must typical that over there is someelement inA the is not an facet ofB. Sowhat would it average to say the the empty collection is not a subsetofB? the would typical that over there is some facet in the empty setthat is not an aspect ofB—but this cannot be true,because there space no facets in the north set! So, indigenous this suggest ofview, it makes some quantity of feeling to say that the empty set is a subset ofevery set.

## Sets the numbers

There space some set of number that space so commonly advantageous that they havespecial symbols. Double-struck funding letters, sometimes referred to as blackboard boldletters, are frequently used (in particular, the letter ℝ,ℤ,ℕ, andℚ). Alternatively, the letters might simply be typeset inboldface.

The set of all actual numbers, both optimistic and an unfavorable (and zero), iscalledR (for “real”). The collection of genuine numbers includesall numbers generally encountered in one algebra, trigonometry, or calculuscourse. (It does no contain facility numbers suchas√−1.)

The set of integers (positive, negative, and zero) is calledZ(from the German indigenous Zahlen, meaning“numbers”). In other words,Z=…,−3,−2,−1,0,1,2,3,….

The collection of herbal numbers is calledN (for“natural”). The set of natural numbers contains all positiveintegers and no negative integers. Unfortunately, over there is no agreement onwhether zero must be taken into consideration a organic number. Some authors include0in the setN, if others carry out not. The reason for this lack ofconsistency is that sometimes it is valuable to incorporate zero and sometimes not,depending top top the situation. For this reason chathamtownfc.netematicians use whichever definition suitsthem best at the time—but lock are mindful of the differences in usage, sothey space always really careful to state precisely which meaning they are usingin any specific case to avoid any type of confusion or ambiguity, and also once castle havedecided top top a meaning they stick v it. Because that this class, let’s agreethat 0is not a organic number unless specified otherwise. Inother words, uneven otherwise stated, us will use the symbolN torepresent the collection 1,2,3,…. If we wish to refer tothe collection 0,1,2,3,… once we are using thisdefinition ofN, we can always write N∪0(we’ll talk about the an interpretation of the symbol∪ in a bit).Alternatively, the terms optimistic integer and also non-negativeinteger are constantly unambiguous: zero is no positive, yet it isnon-negative. Therefore, another method to describe the set1,2,3,… is “the collection of positiveintegers,” while 0,1,2,3,… is “theset that non-negative integers.”

Finally, the set of rational number is calledQ (from the word“quotient”). A reasonable number is a number that can bewritten exactly as a fraction, or quotient, of two integers. For example, thenumber2/3 is a rational number, as is the number−7/2. Allintegers space rational numbers, because any integer can be composed as a fractionwith denominator1; for instance, the integer5 have the right to be writtenas5/1. Other instances of reasonable numbers encompass numbers that have the right to bewritten as a terminating decimal (for example, the number8.13 have the right to bewritten as 813/100) or together a repeating decimal (for example, thenumber0.333… can be created as1/3). Not all genuine numbersare rational, however. Instances of actual numbers that cannot be composed exactlyas a fraction of two integers include√2andπ; the decimal expansions of this numbers go on foreverand never ever repeat. In this course we will not have actually much of a have to distinguishrational numbers from genuine numbers, so us will seldom (if ever) use thesymbolQ.

Note the these 4 sets that numbers room (proper) subsets of each other:NZQR.

## Set-builder notation

Listing all of the facets of a collection is well as long as the collection is no toobig. For bigger sets, we have the right to skip some of the elements by composing anellipsis(…), as we have actually seen, but this is just feasible as soon as theelements follow a pattern that have the right to be plainly seen in the first couple of elements.This is not constantly the case. Because that example, the collection of all prime numbers between100 and500 could be created as101,103,107,…,499, but this is not a veryhelpful point to write due to the fact that it is very daunting to guess: v the correctpattern from these numbers alone (and this expression go not preeminence outincorrect patterns such together “the collection of every odd numbers between 100and500, other than multiples of5”).

In instances like this, it is often much better to describe a collection by specifyinga condition for membership. (Our English description of the collection abovedoes precisely this; that says, in effect, the the problem for a number come anelement the the set is that the number need to be prime and between 100and500.) as soon as we desire to describe a collection in this way, we deserve to useset-builder notation.

When we use set-builder notation, us must very first establish the universalset (sometimes referred to as the domain that discourse or theuniverse of discourse), i m sorry is the collection of all possible objectsunder consideration. Because that example, we might say the the global setisR, the set of all real numbers; or perhapsZ, theset the integers. An additional possibility is to usage a formerly defined collection as theuniversal set. If us have characterized A=1,2,3,for instance, we deserve to use A as the global set.

Once we have actually chosen a universal set, we may “build” a set bychoosing all of the facets of the universal collection that fulfill a specifiedcondition. Because that example, if the universal set isR, we can specify asetB come be, say, the collection of all facets ofR thatare greater than17. (So B consists of the numbers 18and29.4, because that example, but does not contain 11.26 or−30.) ThissetB deserve to be written using set-builder notation as

B=x>17.

In set-builder notation, the upright bar| should be check out as“such that” or “satisfying the problem that.” so theexpression above can be read as “Bis the set thatcontains every elementx in the universal setRsatisfying the condition that x>17.”

Note that the universal collection is stated on the left side of the verticalbar, and also a surname is provided to stand for an arbitrary facet of the universal setby using the symbol∈. Top top the best side that the bar is the conditionthat the element must satisfy in bespeak to it is in a member of the set we arebuilding. The name for the arbitrary element (xin the exampleabove) can be anything; the instance above way exactly the same as, say,

B=z>17,

or even

B=♣>17,

though ♣ is no a really common variable name (and it is likely to causesome quantity of puzzlement, so the should most likely be avoided).

Why is the universal collection important? The prize is that a set described usingset-builder notation will always be a subset that the universal set. Consider theset

C=x>17,

which is defined in exactly the same method as the setB aboveexcept the the universal collection has been changed toN. The setsB andC have many elements in common; for example,the number20 is an element of both B andC.However, because C has been constructed out of facets ofNrather than elements ofR, the setC has only(positive) integers and also does not contain any number with a fractional part. So,for instance, the number23.456 is an facet ofB but isnot an facet ofC.

As one more example the set-builder notation, take into consideration the set

D=n≤10.

What go this mean? analysis from left to right, one symbol in ~ a time, weread, “Dis the set that contains everyelementn in the global setN solve thecondition the n≤10.” Here, the global setis the setN of herbal numbers (which we have actually agreed does notinclude0), so we see that

D=1,2,3,4,5,6,7,8,9,10.

Note the a collection described utilizing set-builder notation might be empty, if noelements in the universal collection satisfy the stated condition! for example, theset

E={n∈N|nF={n∈N|8nn thatsatisfies the inequality 8nnn and also nS to be the set ofperfect squares through writing

S=t∈Z.

(We offered Z as the universal collection here instead ofN, eventhough no an unfavorable integer is a perfect square, because zero is aperfect square and we want to include it in the setS. Thisis an circumstances in which the would have been practically for 0 to beinN. Five well.)

If the universal collection is specified explicitly in indigenous either before or afterthe usage of set-builder notation, the is often omitted in the set-builder notationitself. Because that example, when defining several sets, we might say:

Let

 K = x, L = {x|−6≤xM = x≥50andxisperfect,

where the universal collection is the set of genuine numbers.

In this case, due to the fact that the universal set is specified to it is in the set of realnumbers, we should read the meaning of the setK as“Kis the collection that consists of every realnumberx satisfying the condition thatx≠0.” In various other words, Kis theset that all real numbers except0. Can you understand the definitions ofL andM?

Occasionally a colon(:) is used instead of the vertical bar inset-builder notation. This is especially typical when the condition involves anabsolute worth expression, due to the fact that the upright bars used in the absolute valuenotation have the right to be easily puzzled with the vertical bar offered in the set-buildernotation.

## Set operations

There are several operations that have the right to be perform on set in order to getnew sets from old ones. This operations space as basic to collection theory asaddition and also multiplication space to arithmetic.

### Unions and also intersections

 A = 1,2,3,5,8,13, B = 2,4,6,8,10.

One beneficial thing to be able to do through these sets is to “put themtogether”—in other words, to make a brand-new set that consists of everyelement ofA and additionally every element ofB. Sucha set is referred to as the union that A andB,and is created A∪B. In this example, wehave

A∪B=1,2,3,4,5,6,8,13.

Note the the numbers 2 and8 are each consisted of in both AandB, yet they room each detailed only onceinA∪B, due to the fact that the variety of timesan element is noted in a set is irrelevant.

No matter what the set A andB are, the is alwaystrue thatA⊆(A∪B) andB⊆(A∪B). Doyou check out why?

Another valuable operation is to choose out the aspects that AandB have in common. This is dubbed theintersection that A andB, and is writtenA∩B. In our present example, we have

A∩B=2,8.

No issue what the to adjust A andB are, it is alwaystrue that(A∩B)⊆A and(A∩B)⊆B. Doyou see why?

If the set A andB have no aspects in common,they are claimed to be disjoint. In this instance theintersectionA∩B is the emptyset.

Suppose among the 2 sets we space working through is the empty set. No matterwhat the setA is, the is always true thatA∪∅=A andA∩∅=∅. Execute you see why?

### Complements

Another valuable thing to do with a set is to think about everything the isnot in the set. In bespeak to have actually a precise an interpretation for“everything,” we need to specify a universal set (as we did whenusing set-builder notation). The collection of all aspects of the universal set thatare not elements of a setA is referred to as the complementofA, and also iswrittenA. (Another commonnotation for the match ofAisAc.)

For example, intend the universal collection isN. Allow Pbe the collection of primes; thatis,

P=n∈N=2,3,5,7,11,13,17,….

Then P, the complementofP, is the collection of composite numbers (and the number1,which is neither prime no one composite):

P=1,4,6,8,9,10,12,….

### Set difference

Occasionally we have a should refer to every one of the elements in somesetA that are not elements of part othersetB. This operation is dubbed the set difference(sometimes referred to as the relative complement), and also is writtenAB. (Some authors use a traditional minus signand write A−B, but the procedure of setdifference is quite different from the ordinary idea of subtraction, therefore thebackslash is an ext common.)

For instance, if we have the sets

 A = 3,4,5,10,14,17, B = 4,5,17,

then the set differenceAB is

AB=3,10,14,

because these room the aspects ofA that room not elementsofB.

It is not necessary that either set be a subset that the other. Because that example,consider the sets

 C = 1,3,6,10,15, D = 1,4,5,6,10,20.

Neither of these sets is a subset of the other. The collection difference operationis quiet meaningful, however. Us haveCD=3,15,

because these space the aspects ofC that room not elementsofD; and

DC=4,5,20,

because these room the aspects ofD that are not elementsofC.

Note that, if the universal collection is calledU, then us canexpress the match of a setA as a collection difference:

A=UA.

Let’s think about how the empty collection behaves with the set differenceoperation. No matter what the setA is, it is always true thatA∅=A and∅A=∅. Execute you see why?

### Parentheses

When we do two or an ext set operations, we often need to includeparentheses to make the bespeak of evaluation unambiguous. Because that example, supposewe have the sets

 A = 1,2,3, B = 2,3,4, C = 3,4,5.

Let’s think about the expression(A∪B)∩C. Weevaluate A∪B first, due to the fact that it’s inparentheses. We watch thatA∪B=1,2,3,4,so

(A∪B)∩C=1,2,3,4∩C=3,4.

On the other hand, suppose we have the expressionA∪(B∩C), whichis specifically the same except for the location of the parentheses. Here weevaluate B∩C first, i beg your pardon is3,4, so us get

A∪(B∩C)=A∪3,4=1,2,3,4.

This instance shows that the location of clip is important.Parentheses room free—if you’re unsure around whether friend needparentheses in an expression, placed them in just to be safe.

## Venn diagrams

Venn diagrams, introduced by the English chathamtownfc.netematician john Vennin1881, are really useful for understanding the relationships betweensets.

In a Venn diagram, set are stood for with overlapping shapes. Theoutermost shape, i m sorry is traditionally a rectangle, to represent the universalset. Inside this rectangle are other shapes (often circles) representingvarious sets. This shapes might overlap, denote the opportunity that 2 ormore set may have some aspects in common.

Shown listed below is the most common means to attract a Venn chart for two sets,AandB. Below the universal set iscalledU.

For example, expect the universal collection is

U=1,2,3,…,12

and we define the sets

 A = 1,2,3,7,9,11, B = 3,4,7,11.

The Venn diagram listed below shows the integers indigenous 1 through12 in theappropriate places. Because that instance, the number1 is within the circlelabeledA but outside the one labeledB,because 1 is an element ofA yet not ofB. Thenumber7 is within both the the circles, since it is an facet of bothA andB. The number5 is inside the rectangle,because it is an aspect of the universal set, but it is external both circles,because the is in no A norB.

From this diagram, the is simple to see, for instance, thatA∩B=3,7,11.

A Venn diagram is especially useful when we space thinking around abstract orunknown sets, quite than specific examples such together the sets above. In thiscase we cannot write the facets of the set explicitly; rather we imaginethe various locations of the diagram itself together metaphors for the miscellaneous sets weare functioning with. It is useful to the shade or shade parts the the diagram tohighlight details areas.

The shaded locations in the diagrams listed below show the areas corresponding to theindicated sets.

 A B A∪B A∩B A B AB BA

One usage of Venn diagrams is to verify that 2 expressions actually describethe same set. Because that example, think about the two expressions

(A∪B)(A∩B)and(AB)∪(BA).

Let’s attract Venn diagrams for these sets. We’ll start with theexpression top top the left. The Venn diagrams forA∪B andA∩B are presented below.

 A∪B A∩B

Now the shaded area in the Venn diagram for(A∪B)(A∩B)should include the areas that space shaded in the Venn chart forA∪B yet not shaded in the Venndiagram for A∩B. Therefore the Venn chart for(A∪B)(A∩B)looks like this:

 (A∪B)(A∩B)

Let’s take into consideration the various other expression now:(AB)∪(BA).The Venn diagrams because that AB andBA are displayed below.

 AB BA

Now the shaded area in the Venn diagram for(AB)∪(BA)should encompass the shaded area in the Venn diagram forAB and also the shaded area in the Venndiagram for BA. Therefore the Venn chart for(AB)∪(BA)looks favor this:

 (AB)∪(BA)

Note the this is specifically the very same Venn diagram together we derived for(A∪B)(A∩B).This shows that these two expressions are various ways of specify name the sameset. In various other words, for any type of two sets A andB, the istrue that

(A∪B)(A∩B)=(AB)∪(BA).

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(This collection is sometimes referred to as the symmetric difference ofA andB, writtenAΔB.)