**Equal Sets**

Two sets *A* and *B* are said to be __equal__ if they have exactly the same elements and we write *A*=*B*. Otherwise, the sets are said to be __unequal__ and we write *A*β *B*.

We consider the following examples: (i) Let Y={1,2,3,4} and Z={3,1,4,2}. Then Y=Z.(ii) Let O be the set of prime numbers less than 6 and P the set of prime factors of 30. Then O and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6. |

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Note] A set does not change if one or more elements of the set are repeated. For example, the setsC={1,2,3} andX={2,2,1,3,3} are equal, since each element ofCis inXand viceβversa. That is why we generally do not repeat any element in describing a set.

Definition:

Equal Sets

SetAisequalto setB. symbolized byA=B. if and only if setAand setBcontain exactly the same elements.

For example, if set *D*={1,2,3} and set *E*={3,1,2}, then *D*=*E* because they contain exactly the same elements. The order of the elements in the set is not important. If two sets are equal, both must contain the same number of elements. The number of elements in a set is called its **cardinal number**.

π Set Cardinality β the Number of Elements of a Set

β² Example 1. From the sets given below, select equal sets:

*F*={2,4,8,12}, *G*={1,2,3,4}, *H*={4,8,12,14}, *J*={3,1,4,2}, *K*={-1,1}, *L*={0,a}, *M*={1,-1}, *N*={0,1}

β Solution:

It can be seen that

8β*F*, 8β*G*, 8β*J*, 8β*K*, 8β*L*, 8β*M*, 8β*N*

β *F*β *G*, *F*β *J*, *F*β *K*, *F*β *L*, *F*β *M*, *F*β *N*

Also, 2β*F*, 2β*H*

β΄ *F*β *H*

3β*G*, 3β*H*, 3β*K*, 3β*L*, 3β*M*, 3β*N*

β΄ *G*β *H*, *G*β *K*, *G*β *L*, *G*β *M*, *G*β *N*

12β*H*, 12β*J*, 12β*K*, 12β*L*, 12β*M*, 1β*N*

β΄ *H*β *J*, *H*β *K*, *H*β *L*, *H*β *M*, *H*β *N*

4β*J*, 4β*K*, 4β*L*, 4β*M*, 4β*N*

β΄ *J*β *K*, *J*β *L*, *J*β *M*, *J*β *N*

Similarly, *K*β *L*, *K*β *M*, *K*β *N*, *L*β *M*, *L*β *N*, *M*β *N*.

The order in which the elements of a set are listed is not significant.

β΄ *G*=*J* and *K*=*M*.

Hence, among the given sets, *G*=*J* and *K*=*M*.

β² Ex2. From the following sets, select equal sets.

*O*={2,4,6,8}, *P*={1,2,3,4,5}, *Q*={-2,4,6,8}, *R*={2,3,5,4,1}, *S*={8,6,2,4}.

π Two sets are said to be equal, if they have exactly the same elements.

β Solution:

*O*={2,4,6,8} and *S*={8,6,2,4}

Since, each element of set *O* presents in set *S* and __viceβversa__.

Therefore, *O* and *S* are equal sets.

Similarly, *P*={1,2,3,4,5} and *R*={2,3,5,4,1}

Since, each element of set *P* presents in set *R* and __viceβversa__.

Therefore, *P* and *R* are also equal sets.

β² Ex3. Find the pairs of equal sets, if any, give reasons:

*T*={0}, *V*={*x*:*x*>15 and *x*<5},
*W*={*x*:*x*-5=0}, *Y*={*x*:*x*^{2}=25},

*Z*={*x*:*x* is an integral positive root of the equation *x*^{2}-2*x*-15=0}.

β Solution:

Since 0β*T* and 0 does not belong to any of the sets *V*, *W*, *Y* and *Z*, it follows that, *T*β *V*, *T*β *W*, *T*β *Y*, *T*β *Z*.

Since *V*=Γ but none of the other sets are empty. Therefore *V*β *W*, *V*β *Y* and *V*β *Z*. Also *W*={5} but -5β*Y*, hence *W*β *Y*.

Since *Z*={5}, *W*=*Z*. Further, *Y*={-5,5} and *Z*={5}, we find that, *Y*β *Z*. Thus, the only pair of equal sets is *W* and *Z*.

π Equal sets are a type of sets, so, find other types through this link. Sets and Their Types β Types of Sets