Properties of Relations

We can classify relations based on various properties. These help us understand how the relation behaves and what type of mathematical structure it belongs to.

Reflexive Relation

A relation is reflexive if every element is related to itself.

In other words: every element is connected to itself, so for example (1,1), (2,2), (3,3) are always in the relation.

Example: The "≤" relation is reflexive because every number is less than or equal to itself.

Counterexample: The "<" relation is not reflexive because no number is less than itself.

Symmetric Relation

A relation is symmetric if whenever a is related to b, b is related to a.

In other words: connections are bidirectional; if (a,b) is in the relation, so is (b,a).

Example: The "sibling of" relation is symmetric: if Anna is sibling of Béla, then Béla is sibling of Anna.

Counterexample: The "parent of" relation is not symmetric: if Anna is parent of Béla, Béla is not parent of Anna.

Antisymmetric Relation

A relation is antisymmetric if whenever a is related to b and b to a, then a = b.

In other words: bidirectional connections can only occur between identical elements.

Example: The "≤" relation is antisymmetric: if a ≤ b and b ≤ a, then a = b.

Counterexample: The "friend of" relation is not antisymmetric: if A is friend of B and B of A, they are still different people.

Asymmetric Relation

A relation is asymmetric if whenever a is related to b, b is not related to a.

In other words: no bidirectional connections at all, even between identical elements (strict asymmetry excludes self-loops).

Example: The "<" relation is asymmetric: if a < b, then b < a is false.

Counterexample: The "≤" relation is not asymmetric because if a = b, then (a,b) and (b,a) both hold.

Transitive Relation

A relation is transitive if whenever a is related to b and b to c, then a is related to c.

In other words: connections 'chain': if there's a path of length 2, there's a direct connection.

Example: The "≤" relation is transitive: if a ≤ b and b ≤ c, then a ≤ c.

Counterexample: The "parent of" relation is not always transitive (grandparent cases vary).

Total Relation

A relation is total if every pair of elements is comparable: for any a, b, either (a,b) or (b,a) is in the relation.

In other words: between any two elements, one is related to the other.

Example: The "≤" relation on integers is total because for any two numbers, one is less than or equal to the other.

Counterexample: The divisibility relation is not total because for example 2 and 3, neither divides the other.

Summary

• Reflexive: every element is related to itself.
• Symmetric: if (a,b) ∈ R, then (b,a) ∈ R.
• Antisymmetric: if (a,b) and (b,a) ∈ R, then only if a = b.
• Asymmetric: if (a,b) ∈ R, then (b,a) is definitely not.
• Transitive: if (a,b) and (b,c) ∈ R, then (a,c) ∈ R.
• Total: every two elements are comparable.