# How can this relation be anti-symmetric and symmetric at the same time?

Why is this anti-symmetrical and symmetrical at the same time? I get how it is anti-symmetric because There is no pair such as (1,2) & (2,1) but how did it become symmetrical?

R is a relation on the set of integers
R = {(a,b) | a = b}

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If $R$ is some relation that is both antisymmetric and symmetric, what can you conclude about $R$? – wj32 Oct 7 '12 at 4:05
But how can it be symmetric when there is no pair where (a,b) & (b,a) are in the Relation where a is not equal to b? – Maximus Programus Oct 7 '12 at 4:07
What is your definition of symmetric? – wj32 Oct 7 '12 at 4:08
It's symmetric because whenever $(a,b)\in R$, then $(b,a)$ must be in $R$ also. In fact, any other relation on the integers that is both symmetric and anti-symmetric must be a subset of this one. – user22805 Oct 7 '12 at 4:08
You may wish to check your definition of symmetric. You may have misunderstood it. – user22805 Oct 7 '12 at 4:14

If $(a,b)$ is in $R$ then $a=b$ [by the definition given for $R$] so $(b,a)=(a,b)$ so $(b,a)$ is in $R$.