Equal Sets

Equal Sets. Two sets, A and B, are considered equal if they consist of the exact same elements. This relationship is denoted by the equation A = B.

For example, set A includes the elements {1, 2, 3, 4, 5, 6}, and set B contains {1, 2, 3, 4, 5, 6}. Since both sets comprise the same elements, we can confirm that they are indeed equal. Below is a graphical representation of equal sets:

If two sets A and C are not equal, this is indicated by A≠C. Consider, for instance, set A={1, 2, 3, 4, 5, 6} and set C={5, 6, 7, 8, 9}. These sets do not share all the same elements, making them distinct and not equal.

Equal sets exhibit a mutual double inclusion. Sets A and B are equal if and only if set A is included in set B and set B is included in set A. In other words, "two sets are equal if every element of the first set is also found in the second, and vice versa".

A = B ⇔ A ⊆ B and A ⊇ B

The converse is also true. A condition of mutual double inclusion occurs if and only if the two sets are equal ( set equality ).

Example 1

Sets A and B are equal sets

$$ A = \{ 1,3,5,7 \} $$

$$ B = \{ 5,1,7,3 \} $$

Each element of set A is in set B. Each element of set B is in set A

Example 2

Sets A and C are not equal sets.

$$ A = \{ 1,3,5,7 \} $$

$$ C = \{ 5,1,7,4 \} $$

The element 3 is in the set A but not in the set C. The element 4 is in C but not in A.