Question

Determine whether the relation o defined on ℤ is reflexive, symmetric, or transitive. The relation o on ℤ is defined as follows: for all m, n ∈ ℤ, m o n ⇔ m – n is odd

Answer 1

• Reflexive: no. A relation is reflexive is an element is in relation with itself. But in this case, it cannot happen, because $x \circ x \iff x-x \text{ is odd} \iff 0 \text{is odd}$, which clearly cannot happen
• Symmetric: yes. A relation is symmetric if every time x and y are in relation, then also y and x are in relation. In this case, you have x and y are in relation if x-y is odd. But then, this guarantees that y and x are in relation, because $y-x = -(x-y)$, and the opposite of an odd number is still odd
• Transitive: no. A relation is transitive if every time x and y are in relation, and y and z are in relation, then x and z are in relation. So, suppose that x and y are in relation, which means $x-y = 2k+1$, for some integer k. We also know that y and z are in relation, which means that $x-y = 2m+1$, for some integer m. But then, you have

$x-z = x-y+y-z = 2k+1 + 2m+1 = 2k+2m+2 = 2(k+m+1)$

But since $2(k+m+1)$ is twice some integer, it is even, and thus x and z are not in relation. So, we've proven that although $x \circ y$ and $y \circ z$, it can't be that $x \circ z$, and thus the relation is not transitive.