Equivalent Relations

Question 1

What is an equivalent relation?

Answer 1

An equivalent relation is a relation on a set X that is reflexive, symmetric, and transitive.

Question 2

What does it mean for a relation to be transitive?

Answer 2

A relation R is transitive if (a, b) ∈ R, and (b, c) ∈ R, implies (a, c) ∈ R.

Or, if aRb and bRc implies (=>) aRc

Question 3

What is meant by a a relation that is symmetric?

Answer 3

A relation R is said to be symmetric if the ordered pair (a, b) ∈ R => (b, a) ∈ R.

Question 4

What is meant by a relation that is reflexive?

Answer 4

A relation R is said to be reflexive if (a, a) ∈ R. That is, if the relation aRa exists.

Question 5

What is a relation?

Answer 5

A relation is a way/method of describing the connection between elements of the given sets. It is also a subset of the Cartesian product of the given sets.

Question 6

What is meant by “pairwise disjoint”?

Answer 6

A collection of sets is said to be pairwise disjointed if no two sets in the collection share a common element, or if the intersection of the sets is an empty set.

Question 7

What is an equivalent class?

Answer 7

An equivalent class is a subset of X under a given equivalent relation, whose elements are equivalent to each other.