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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 $R$?2012-10-07
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    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?2012-10-07
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    What is your definition of symmetric?2012-10-07
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    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.2012-10-07
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    @wj32 a relation is symmetric if (a,b) & (b,a) are in the relation where a is not equal to b, a relation is antisymmetric if there are no distinct a and b with (a,b) & (b,a) in the relation2012-10-07
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    @DavidWallace but then it would imply that (a = b)2012-10-07
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    You may wish to check your definition of symmetric. You may have misunderstood it.2012-10-07

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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 $R$.