Types of Relations: Reflexive, Symmetric, Transitive Reflexive Relation: A relation R on a set S is reflexive if, for all elements a, b in S, if a R b,...
Types of Relations: Reflexive, Symmetric, Transitive
Reflexive Relation:
A relation R on a set S is reflexive if, for all elements a, b in S, if a R b, then b R a.
Examples:
The relation "being equal to" on the set of real numbers is reflexive.
The relation "is sibling to" on the set of people is reflexive.
Symmetric Relation:
A relation R on a set S is symmetric if, for all elements a, b in S, if a R b, then b R a.
Examples:
The relation "is congruent to" on the set of shapes is symmetric.
The relation "is divisible by" on the set of natural numbers is symmetric.
Transitive Relation:
A relation R on a set S is transitive if, for all elements a, b, c in S, if a R b and b R c, then a R c.
Examples:
The relation "is greater than" on the set of real numbers is transitive.
The relation "is contained in" on the set of rooms in a school is transitive.
Additional Notes:
A relation can be reflexive, symmetric, or transitive, but not all relations are of these types.
A relation that is both reflexive and symmetric is called an equivalence relation.
A relation that is transitive and reflexive is called a partial order