# Equivalent Sets — One-to-One Correspondence among Members cross Two Sets

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

Definition: Equivalent Sets
Set A is equivalent to set B if 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 sets D={a,b,c} and E={apple, orange, pear} are equivalent because both have the same cardinal number, 3. Because the elements differ, however. the sets are not equal.

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.

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

H={1,2,3}, I={t,p,q,r,s}, J={α,β,γ}, and K={a,e,i,o,u}

⛲ 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}

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

⛲ Ex3. State whether the following pairs of sets are equivalent or not:
(i) V= {x:x∈ℕ and 11≥2x-1} and
W={y:y∈𝕎 and 3≤y≤9}
(ii) Set of whole numbers and set of multiples of 3.
(iii) X={5,6,7,8} and Y={x:x∈𝕎 and x≤4}
✍ Solution:
(i) V={x:x∈ℕ and 11≥2x-1}

 11≥2x-1 11+1≥2x-1+1 (Adding 1 to both sides) 12÷2≥2x÷2 6≥x V={1,2,3,4,5,6} |V|=n(V)=6

W= {y:y∈𝕎 and 3≤y≤9}

 3≤y≤9 W={3,4,5,6,7,8,9} |W|=n(W)=7

∴ 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)=4 Y={x:x∈𝕎 and x≤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.
🌈 Set Cardinality — the Number of Elements of a Set