Equal Sets

Two sets, A and B, are deemed equal or identical sets if they exactly match in elements.

Mathematically, this holds when each element of A is an element of B, and each element of B is also an element of A.

This leads to the conclusion that \( A \subseteq B \) (A is a subset of B) and \( B \subseteq A \) (B is a subset of A).

$$ A \subseteq B \wedge B \subseteq A $$

To state that two sets are identical more succinctly, we write \( A = B \).

$$ A = B $$

This declaration means that not only is every element of A found in B, but there are also no elements unique to either set.

Thus, these statements are equivalent:

$$ A = B \Leftrightarrow A \subseteq B \wedge B \subseteq A $$

When two sets are equal, they naturally share identical elements and thus also possess the same cardinality—that is, they have the same number of elements. $$ |A| = |B| $$

Example

For a practical example, consider sets A and B:

$$ A = \{1, 2, 3\} $$

$$ B = \{3, 1, 2\} $$

Although the order of the elements differs, sets A and B consist of the identical numbers.

Thus, \( A = B \) holds true because both sets precisely include the same elements,