Suppose set X = {1, 2, 3}. I'll define the relation R = {(1,3), (3, 1)}. Is R symmetric or not.

Question

I would assume R is symmetric because 1, 3 and 3, 1 are swapped, but in the definition of a symmetric relation, the property must hold for all a,b in the set X. But 2 is not covered in the relation.

Answer 1

The fact that 2 is not covered by the relation $R$ has no impact on the symmetry of $R$. The symmetry property doesn't say on its own whether any elements are related, only that each related pair is reversible.

If you want to be careful about it, you can make a table of all the pairs $(a,b)$ from $X\times X$ and verify that $a \mathrel{R} b \iff b \mathrel{R} a$ is satisfied:

a b aRb bRa (aRb <=> bRa)
-------------------------
1 1  F   F        T
1 2  F   F        T
1 3  T   T        T
2 1  F   F        T
2 2  F   F        T
2 3  F   F        T
3 1  T   T        T
3 2  F   F        T
3 3  F   F        T