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A relation $R$ on a set $A$ is called symmetric if$(b,a) ∈ R$ whenever $(a,b) ∈ R$,for all $a,b ∈ A$. A relation $R$ on a set $A$ such that for all $a, b ∈ A$, if $(a, b) ∈ R$ and $(b, a) ∈ R$, then $a = b$ is called antisymmetric.

Shouldn't it be $a≠b$ for antisymmetric? This is on my textbook and I consider it a errata, but I can't find this from the official errata list. please let me know your thought.

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The book is correct, it meant to say that whenever $(a,b)\in R$ then the reverse pair $(b,a)\in R$ if and only if $a=b$. However you may like the alternate statement:

$\forall a,b\in A$, if $(a,b)\in R$ and $a\ne b$ then $(b,a)\notin R$

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    It might help the OP if you add that the alternative statement is logically equivalent to the book's statement. Your alternate doesn't change the meaning but only how it's expressed. – 2017-01-21
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    I have edited it. Let me know what value it adds to the solution of OP. – 2017-01-21
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    My concern was that, in view of the level of the question, the equivalence of the two formulations might not be evident to the OP, who might incorrectly think that you are proposing a different notion of antisymmetry. – 2017-01-21
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    Yes..you were right..thanks for pointing out. I have tried to improve my answer. See my edit. – 2017-01-22
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Think about $\leq$, which is antisymmetric. If $a\leq b$ and $b \leq a$, then we must have $a = b$. At the end of the day, we need a name for the property, and since the property says "If $a$ and $b$ are unequal, then we never have both $(a, b)\in R$ and $(b, a)\in R$", in other words, the "symmetry" condition is never true (except for elements that are equal), antisymmetric doesn't sound too bad to me.