Question

Amongst the properties {reflexivity, symmetry, antisymmetry, transitivity} the relation R={(a,b) ∈ N^2 | a!= b} satisfies _______ property.

(a) symmetry

(b) transitivity

(c) antisymmetry

(d) reflexivity

The question was asked by my college professor while I was bunking the class.

The query is from Closure on Relations topic in division Relations of Discrete Mathematics

Answer 1

Right choice is (a) symmetry

To elaborate: It is not reflexive as aRa is not possible. It is symmetric as if aRb then bRa. It is not antisymmetric as aRb and bRa are possible and we can have a!=b. It is not transitive as if aRb and bRc then aRc need not be true. This is violated when c=a. So the answer is symmetry property.