Two sets are said to be equivalent if they contain the same number of elements.

Definition:

Equivalent Sets

SetAisequivalentto setBif and only if n(A)=n(B).

Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal, however. The setsD={a,b,c} andE={apple, orange, pear} are equivalent because both have the same cardinal number, 3. Because the elements differ, however. the sets are not equal.

π Equal Sets β What are Equal Sets?

Two sets that are equivalent or have the same cardinality can be placed in **one-to-one correspondence.** Set *A* and set *B* can be placed in one-to-one correspondence if every element of set *A* can be matched with exactly one element of set *B* and every element of set *B* can be matched with exactly one element of set *A*. For example, there is a one-to-one correspondence between the student names on a class list and the student identification numbers because we can match each student with a student identification number.

Consider set *S*, states. and set *C*, state capitals.

*S*={North Carolina, Georgia, South Carolina, Florida}

*C*={Columbia, Raleigh, Tallahassee, Atlanta}

Two different one-to-one correspondences for sets *S* and *C* follow

Other one-to-one correspondences between sets *S* and *C* are possible. Do you know which capital goes with which state?

**Equivalent Sets**

Two sets *A* and *B* are equivalent, if their cardinal numbers are same i.e., n(*A*)=n(*B*).

e.g., Let *F*={a,b,c,d} and *G*={1,2,3,4}, then n(*F*)=4 and n(*G*)=4.

Therefore, *F* and *G* are equivalent sets.

β² Example 1. From the sets given below, pair the equivalent sets.

*H*={1,2,3}, *I*={t,p,q,r,s}, *J*={Ξ±,Ξ²,Ξ³}, and *K*={a,e,i,o,u}

π Answer: *A/em>, J<; I,K*

β² Ex2. From the sets given below, select equal sets and equivalent sets.

L={0,a}, M={1,2,3,4}, N={4,8,12}, O={3,1,2,4}, P={1,0}, Q={8,4,12}, R={1,5,7,11}, and T={a,b} |

π Answer:

Equal sets-*M*=*O*,*N*=*Q*, and Equivalent sets-*L*,*P*,*T*; *M*,*O*,*R*; *N*,*Q*

β² Ex3. State whether the following pairs of sets are equivalent or not:

(i) *V*= {*x*:*x*ββ and 11β₯2*x*-1} and

*W*={*y*:*y*βπ and 3β€*y*β€9}

(ii) Set of whole numbers and set of multiples of 3.

(iii) *X*={5,6,7,8} and *Y*={*x*:*x*βπ and *x*β€4}

β Solution:

(i) *V*={*x*:*x*ββ and 11β₯2*x*-1}

11β₯2x-111+1β₯2 x-1+1 (Adding 1 to both sides)12Γ·2β₯2 xΓ·26β₯x V={1,2,3,4,5,6}| V|=n(V)=6 |

*W*= {*y*:*y*βπ and 3β€*y*β€9}

3β€yβ€9W={3,4,5,6,7,8,9}| W|=n(W)=7 |

β΄ Cardinal number of set *V*=6 and cardinal number of set *W*=7.

Hence, set *V* and set *W* are not equivalent.

(ii) Set of whole numbers and set of multiples of 3 are equivalent because both these sets have infinite number of elements.

(iii) *X*={5,6,7,8}

n(X)=4Y={x:xβπ and xβ€4}Y={0,1,2,3,4}n( Y)=5 |

Cardinal number of set *X*=4 and Cardinal number of set *Y*=5.

Hence, these sets are not equivalent.

π Set Cardinality β the Number of Elements of a Set

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