Equal Sets and Equivalent Sets

Consider two sets:

A = {−9, −3, 0, 5, 12}

B = {−2, 1, 2, 4, 7}

Did you notice any relation between the sets A and B?

Let us see.

We have,

A = {−9, −3, 0, 5, 12}

B = {−2, 1, 2, 4, 7}

Therefore, we have n (A) = 5 and n (B) = 5

Observe that both the sets A and B have same number of elements. Therefore, in this case, we say that the sets A and B are equivalent sets and it can be defined as:

Two finite sets are called equivalent, if they have the same number of elements.

Thus, two finite sets X and Y are equivalent, if n (X) = n (Y). We write it as X ↔ Y (read as “X is equivalent to Y”)

Now, consider the two sets:

X = {all letters in the word STONE}

Y = {all letters in the word NOTES}

Did you notice any relation between the sets X and Y?

Let us see.

We have,

X = {S, T, O, N, E} and Y = {N, O, T, E, S}

Observe that both the sets X and Y have same elements. Therefore, in this case, we say that the sets X andY are equal sets.

Two sets are called equal, if they have same elements.

When two sets X and Y are equal, we denote it as X = Y; and if they are not equal, then we write it as X ≠Y

Also, note that n (X) = 5 and n (Y) = 5

Therefore, we can conclude that:

If A and B are finite sets and A = B, then n (A) = n (B) i.e., A and B are equivalent. However, the converse of the above statement may not be true.

For example, if A = {2, 4, 6} and B = {1, 3, 5}, then n (A) = n (B) = 3; however, A ≠ B

Let us now look at some more examples to understand the above discussed concepts better.

Example 1:

Which of the following sets are equal?

(a) X = {x: x is a letter in the word REFRESH},

Y = {A letter in the word FRESHER}

(b) X = {4}, Y = {x: x∈ N, x − 4 = 0}

(d) X = {x/x is a vowel letter in the word WEIGHT},

Y = {x/x is a vowel letter in the word HEIGHT}

(c) X = {x: x∈ N, 0 < x < 4}, Y = {x: x∈ W}

Solution:

(a) X = {R, E, F, S, H}, Y = {F, R, E, S, H}

∴ X and Y are equal sets i.e., X = Y

(b) X = {4}, Y = {4}

∴ X and Y are equal sets i.e., X = Y

(c) X = {E, 1}, Y = {E, I}

∴ X and Y are equal sets i.e., X = Y

(d) X = {1, 2, 3}, Y = {0, 1, 2, 3, 4, 5 …}

∴ X and Y are not equal sets i.e., X ≠ Y

Example 2:

Which of the following sets are equal?

(a) X = {x: x is a vowel in the word MATRIX}

Y = {A vowel in the word SHIVANI}

(b) X = {5, 0, 1, 0, 2, 1}, Y = {3, 2, 8, 3, 3, 2}

Solution:

(a) X = {A, I}, Y = {A, I}

∴ n (X) = 2 and n (Y) = 2

This means n (X) = n (Y)

Therefore, X and Y are equivalent sets.

(b) X = {5, 0, 1, 0, 2, 1} = {0, 1, 2, 5}, Y = {3, 2, 8, 3, 3, 2} = {2, 3, 8}

∴ n (X) = 4 and n (Y) = 3

This means n (X) ≠ n (Y)

Therefore, X and Y are not equivalent sets.