Question

State whether the following pair of sets are equivalent but not equal.

$\text{A =}\left\{1,2,3,4,........\right\}$, $\text{B =}\left\{0,1,2,3,.......\right\}$

Answer 1

Step $\mathbf{1}$: Definitions:

Equal sets : Two sets are said to be equal if they have the same elements. All equal sets have equal number of elements. But all sets having equal number of elements may or may not be equal.

Equivalent sets: Two sets are said to be equivalent if the number of elements in both the sets are equal. All equal sets are equivalent but equivalent sets may or may not be equal.

Step $\mathbf{2}$: Explanation that the sets are equivalent but not equal.

Here,

$\text{A =}\left\{1,2,3,4,........\right\}$

and

$\text{B =}\left\{0,1,2,3,.......\right\}$

Here, number of elements in set A $=\infty$ and number of elements in set B $=\infty$.

As, the numbers of elements are same but elements are different.

So, the set A and set B are equivalent but not equal.

Hence, the given pair of sets are equivalent but not equal.