Relation

• Suppose is a non-empty set. A binary relation is an ordered pair of two elements of the set. This is denoted as or

Example: Our set is , relation = divisibility by 3. We will say if Elements related with 3 are , which are natural numbers divisible by 3. Example: Define Then,

• Given that is a non-empty set, is a binary relation, is reflexive if for all .
• A relation is symmetric if and are equal, for all in
  • A trivial example of a symmetric relation is equivalence ()
• If a relation is transitive, then if and then , for all in

Example: is an example of a transitive relation. Example: In , is a a binary relation defined as is even. This relation is not transitive because is 6 which is even, and is 10 which is even, but is 15 which is not even

• A relation on a non-empty set is called an equivalence relation if is reflexive, symmetric, and transitive.

Example: Consider two sets, . Relate each element of to a unique element of

• A nicer way to describe relations is a Functions.