Relations

A relation on a set is a subset of the Cartesian product of the set with itself. In this case, the relation is {(1, 2), (2, 3), (3, 1)} because it is a subset of the Cartesian product {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.

A reflexive relation is a relation where every element in the set is related to itself. In this case, the relation {(1, 1), (2, 2), (3, 3)} is reflexive because every element in the set {1, 2, 3} is related to itself.

A symmetric relation is a relation where if (a, b) is in the relation, then (b, a) is also in the relation. In this case, the relation {(1, 2), (2, 1), (3, 3)} is symmetric because if (1, 2) is in the relation, then (2, 1) is also in the relation.

A transitive relation is a relation where if (a, b) and (b, c) are in the relation, then (a, c) is also in the relation. In this case, the relation {(1, 2), (2, 3), (3, 1)} is transitive because if (1, 2) and (2, 3) are in the relation, then (1, 3) is also in the relation.

An equivalence relation is a relation that is reflexive, symmetric, and transitive. In this case, the relation {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)} is an equivalence relation because it is reflexive, symmetric, and transitive.

The domain of a relation is the set of all first components of the ordered pairs in the relation. In this case, the domain of the relation {(1, 2), (2, 3), (3, 1)} is {1, 2, 3} because the first components of the ordered pairs are 1, 2, and 3.

The range of a relation is the set of all second components of the ordered pairs in the relation. In this case, the range of the relation {(1, 2), (2, 3), (3, 1)} is {1, 2, 3} because the second components of the ordered pairs are 1, 2, and 3.

The inverse of a relation is the relation that consists of all the ordered pairs (b, a) such that (a, b) is in the original relation. In this case, the inverse of the relation {(1, 2), (2, 3), (3, 1)} is {(2, 1), (3, 2), (1, 3)}.

The composition of two relations R and S is the relation that consists of all the ordered pairs (a, c) such that there exists an element b such that (a, b) is in R and (b, c) is in S. In this case, the composition of the relations {(1, 2), (2, 3)} and {(3, 4), (4, 5)} is {(1, 4), (2, 5)}.

The union of two relations R and S is the relation that consists of all the ordered pairs that are in either R or S. In this case, the union of the relations {(1, 2), (2, 3)} and {(3, 4), (4, 5)} is {(1, 2), (2, 3), (3, 4), (4, 5)}.

The intersection of two relations R and S is the relation that consists of all the ordered pairs that are in both R and S. In this case, the intersection of the relations {(1, 2), (2, 3)} and {(3, 4), (4, 5)} is {} because there are no ordered pairs that are in both relations.

The difference of two relations R and S is the relation that consists of all the ordered pairs that are in R but not in S. In this case, the difference of the relations {(1, 2), (2, 3)} and {(3, 4), (4, 5)} is {(1, 2), (2, 3)} because the ordered pairs (1, 2) and (2, 3) are in R but not in S.

The symmetric difference of two relations R and S is the relation that consists of all the ordered pairs that are in either R or S but not in both. In this case, the symmetric difference of the relations {(1, 2), (2, 3)} and {(3, 4), (4, 5)} is {(1, 2), (2, 3), (3, 4), (4, 5)} because the ordered pairs (1, 2), (2, 3), (3, 4), and (4, 5) are in either R or S but not in both.

The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) such that a is in A and b is in B. In this case, the Cartesian product of the sets {1, 2} and {3, 4} is {(1, 3), (1, 4), (2, 3), (2, 4)}.