Given any $(a_i,a_j)\in A\times A$ and if R is anti-symmetric relation, we have three cases:
1. $a_i R a_j$ and $a_j Ra_i$ so that $a_i Ra_i$
2. $a_i R a_j$ and $(a_j, a_i)\notin R$.
3. $a_j Ra_i$ and $(a_i, a_j)\notin R$
4. $(a_i,a_j)\notin R$ and $(a_j,a_i)\notin R$