### Theory:

If two finite sets \(A\) and \(B\) have the same number of elements, then they are called equivalent sets. It is denoted by $A\approx B$. If \(A\) and \(B\) are equivalent sets, then \(n(A) = n(B)\).

Example:

**1.**Consider the set $A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\{c,d\}$ and $B=\phantom{\rule{0.147em}{0ex}}\{1,2\}$.

Here the sets \(A\) and \(B\) are equivalent sets as \(n(A) = n(B) = 2\).

**2.**Consider the set $A=\phantom{\rule{0.147em}{0ex}}\{x:x\phantom{\rule{0.147em}{0ex}}\in \mathrm{\mathbb{N}},2<x<10\}$ and the set of all months in a year having \(31\) days.

Here the set \(A\) belongs to natural number between \(2\) and \(10\).

That is, $A=\phantom{\rule{0.147em}{0ex}}\{3,4,5,6,7,8,9\}$.

And let the set of all months in a year having \(31\) days be \(B\).

The months having 31 days in a year are January, March, May, July, August, October and December. That is, $B=\phantom{\rule{0.147em}{0ex}}\{\mathit{January},\phantom{\rule{0.147em}{0ex}}\mathit{March},\phantom{\rule{0.147em}{0ex}}\mathit{May},\phantom{\rule{0.147em}{0ex}}\mathit{July},\phantom{\rule{0.147em}{0ex}}\mathit{August},\phantom{\rule{0.147em}{0ex}}\mathit{October},\phantom{\rule{0.147em}{0ex}}\mathit{December}\}$.

The sets \(A\) and \(B\) has \(6\) elements in it. Thus, \(n(A) = n(B) = 6\).

If two sets \(A\) and \(B\) are equal sets, then they contain exactly the same elements; otherwise, they are said to be unequal.

- For equal sets, each element of set \(A\) belongs to set \(B\) and vice versa.
- All equal sets are equivalent sets, but all equivalent sets are need not be equal sets.

Important!

Example:

Consider the set $A=\phantom{\rule{0.147em}{0ex}}\{\mathit{Red},\mathit{Orange},\mathit{Yellow},\mathit{Green},\mathit{Blue},\mathit{Indigo},\mathit{Violet}\}$. Another set contains the set of all colour codes in the rainbow.

Here the set \(A\) is given in roaster form. Let us express the set \(B\) also in roaster form.

The colour codes of rainbow follow VIBGYOR. That is Violet, Indigo, Blue, Green, Yellow, Orange and Red.

Thus, $B=\phantom{\rule{0.147em}{0ex}}\{\mathit{Violet},\mathit{Indigo},\mathit{Blue},\mathit{Green},\mathit{Yellow},\mathit{Orange},\mathit{Red}\}$.

Therefore, both sets \(A\) and \(B\) have the same elements. So they are equal sets.

- If the sets \(A\) and \(B\) are equal, then \(A=B\).
- If the sets \(A\) and \(B\) are unequal, then \(A≠B\).

Important!