Set Equality In Mathematics: Definition And Examples

Equality of Sets

Two sets are equal if they contain exactly the same elements.

Two sets are said to be equal if they contain exactly the same elements. That is, if A and B are two sets, then A = B if and only if every element of A is an element of B and every element of B is an element of A. This can be expressed mathematically as:

A = B if and only if (x ∈ A ⇔ x ∈ B)

where x is an arbitrary element.

For example, if A = {1, 2, 3} and B = {2, 3, 1}, then A = B because they contain the same elements, although they are listed in a different order. However, if C = {1, 2, 3, 4} then A and C are not equal because A does not contain the element 4.

It is important to note that the order of the elements in a set does not matter. Therefore, {1, 2, 3} is the same as {3, 2, 1} and {1, 1, 2, 3} is the same as {1, 2, 3}.

In order to show that two sets are equal, it is often sufficient to show that each set is a subset of the other. That is, if A ⊆ B and B ⊆ A, then A = B. Conversely, if one can show that two sets are not equal, it is enough to find a single element that belongs to one set but not the other.

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