Frequently Asked Questions
An ordered pair is a pair of elements written as \((a, b)\), where the order matters. Two ordered pairs are equal if and only if their corresponding elements are equal.
The Cartesian product of sets \(A\) and \(B\), denoted \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\).
If set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, then \(A \times B\) has \(m \times n\) elements.
A relation from set \(A\) to set \(B\) is any subset of the Cartesian product \(A \times B\).
The domain is the set of all first elements of the ordered pairs belonging to the relation.
The range is the set of all second elements of the ordered pairs of a relation.
The codomain is the set from which the second elements of ordered pairs are taken, regardless of whether all elements appear in the relation or not.
A relation that contains no ordered pair is called an empty relation.
A relation that contains all possible ordered pairs of a Cartesian product is called a universal relation.
An identity relation on a set \(A\) consists of all ordered pairs \((a, a)\) for every \(a \in A\).
A relation is reflexive if every element of the set is related to itself, i.e., \((a, a)\) belongs to the relation for all \(a\).
A relation is symmetric if whenever \((a, b)\) belongs to the relation, \((b, a)\) also belongs to it.
A relation is transitive if whenever \((a, b)\) and \((b, c)\) belong to the relation, then \((a, c)\) must also belong to it.
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
An equivalence class is the set of all elements related to a given element under an equivalence relation.