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.

🌈 Equal Sets — 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

one-to-one correspondence

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}
🔑 Answer: A/em>,J<; I,K

⛲ 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}

🔑 Answer:
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

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