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

Reflexive: no. A relation is reflexive is an element is in relation with itself. But in this case, it cannot happen, because , which clearly cannot happenSymmetric: 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 , and the opposite of an odd number is still oddTransitive: 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 , for some integer k. We also know that y and z are in relation, which means that , for some integer m. But then, you have

But since is twice some integer, it is even, and thus x and z are not in relation. So, we've proven that although and , it can't be that , and thus the relation is not transitive.