Question

What is set equivalence?

Answer 1

Set equivalence refers to a relationship between two sets where they contain exactly the same elements. Two sets \( A \) and \( B \) are considered equivalent (or equal) if every element of set \( A \) is also an element of set \( B \), and every element of set \( B \) is also an element of set \( A \). This can be expressed formally as:

\[ A = B \iff (A \subseteq B) \land (B \subseteq A) \]

Here, \( A \subseteq B \) means that \( A \) is a subset of \( B \), and \( B \subseteq A \) means that \( B \) is a subset of \( A \).

In other words, if two sets are equivalent, they have the same cardinality (number of elements) and consist of the same members, regardless of the order of the elements or any repetition (since sets do not allow duplicate entries). For example, the sets \( \{1, 2, 3\} \) and \( \{3, 2, 1\} \) are equivalent because they contain the same elements. Conversely, the sets \( \{1, 2\} \) and \( \{1, 2, 3\} \) are not equivalent because they do not contain the same elements.