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Antisymmetric:

if R(a,b) and R(b,a), then a = b, or, equivalently, if R(a,b) with a ≠ b, then R(b,a) must not hold.

If a relation is irreflexive as well as transitive then it should be asymmetric and antisymmetric

for example:

A={1,2,3}

R={(1,2)(2,3),(1,3)} is irreflexive as well as transitive

and it is asymmetric and antisymmetric

if a relation is asymmetric then it must be antisymmetric also

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Anti-symmetric and Asymmetric are 2 different properties

A relation R on set A such that for all a,b $\in$ A, if (a,b) $\in$ R and (b,a) $\in$ R, then a = b is called antisymmetric

A relation R on set A such that all a and b in A, if a is related to b, then b is not related to a is called Asymmetric. 


  • A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
  • A transitive relation is asymmetric if and only if it is irreflexive.
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option a and b both are true.

S=1,2,3

R=$\left \{ 1,2 \right \}$ it is irreflexive and transitive .

and it ia also antisymmetric and asymmetric

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