Symmetric AntiSymmetric Asymmetric Relations

In this class, We discuss Symmetric AntiSymmetric Asymmetric Relations.

The reader should have prior knowledge of reflexive property. Click Here.

Symmetric Relation:

A relation R is said to be symmetric if xRy then yRx ∀(x,y) ∈ R

Example:

A = {1, 2, 5}

R1 = {(1, 2), (2, 1), (1, 5), (5, 1), (1,1)}

The relation R1 is symmetric. because for all (x, y) pairs we have (y, x) pairs in the relation.

R2 = {(1, 2), (1, 5), (5, 1)}

The relation R2 is not symmetric. Because we do not have ordered pair (1, 2).

R3 = {}

empty set is a symmetric relation. Because we need to check for available (x, y) pairs.

Anti Symmetric:

A relation is considered anti-symmetric if xRy and yRx, then x=y ∀(x,y) ∈ R.

A = {1, 2, 5}

R1 = {(1, 1), (2, 2)}

The above relation is anti-symmetric.

R2 = {(1, 2), (2, 1)}

Relation R2 is not an anti-symmetric.

R3 = {(1, 2), (1, 1)}

Relation R3 is an anti-symmetric relation

Asymmetric Relation:

A relation is said to be asymmetric if xRy, then y should not relate x ∀(x,y) ∈ R.

A = {1, 2, 5}

R1 = {(1, 2), (1,5)}

Relation R1 is a symmetric relation.

R2 = {(1, 2), (1, 1)}

Relation R2 is not an asymmetric relation

R3 = {(1,2), (2, 1)}

Relation R3 is not an asymmetric relation.

R4 = {}

Relation R4 is an asymmetric relation.