Is this relation symmetric, anti-symmetric or neither?

Question

The relation R on the set ℤ is defined by the rule R = {(x,y) ∈ ℤ : xy+y is even}.

For example: (5,3) ∈ ℤ, and (3,5)∈ ℤ, both would be even so this would be symmetric.

For a counterexample: (3,4) ∈ ℤ but (4,3) would not be an element of the set since 4(3)+3= 15, so this example would be anti-symmetric. Would that mean neither is the answer since it has symmetric and anti-symmetric examples?

Answer 1

Yes, it would mean that it is neither. You've found one counterexample pair $(5,3)$ which shows that the relation isn't antisymmetric and one pair $(4,3)$ which shows it isn't symmetric.

Don't think of a single pair like $(5,3)$ as symmetric and $(4,3)$ as antisymmetric. A whole relation can have such a property, but not a single pair (or a pair of pairs). A single pair can, however, disprove that a relation has such a property, like in this case.

Answer 2

It is not symmetric because

$$(3,4)\in R \text{ but } (4,3)\notin R$$

it is not antisymmetric because

$$(2,4)\in R \text{ and } (4,2)\in R \text{ and } 2\ne 4$$