# Determing whether or not the relationships in each problem are symmetric, transitive, and/or reflexive

For each of the following relations on the set of all integers, determine whether the relation is reflexive, symmetric, and/or transitive:

a. (𝑥,𝑦)∈𝑅 if and only if 𝑥<𝑦.

b. (𝑥,𝑦)∈𝑆 if and only 𝑥𝑦≥1.

c. (𝑥,𝑦)∈𝑇 if and only 𝑥=−𝑦.

d. (𝑥,𝑦)∈𝑈 if and only 𝑥=|𝑦|

I've made some attempts at solving these of course, but have only yielded fitting answers for b and d, which I know are symmetric/transitive, and solely symmetric, respectively (unless I made some errors in deducing this which is definitely not out of the question). I even tried looking at post like the one below to determine whether or not what I was doing was correct, but it didn't provide much I could really use.
I appreciate your help and assistance

Determine whether the relations are symmetric, antisymmetric, or reflexive.

• Welcome to Mathematics Stack Exchange. For reflexive, for all $x$, is (a) $x<x$? (b) $x^2\ge1$? (c) $x=-x$? (d) $x=|x|$? – J. W. Tanner May 3 at 19:12

In this type of exercise, you need to apply the definition of these types of relations and you have to ask yourself the right questions:

• Reflexivity : for all $$x$$, is it true that $$x < x \text{ (a). }\\ x^2 \ge 1 \text{ (b). }\\ x = -x \text{ (c). }\\ x = |x| \text{ (d). }$$

• Symmetry : for all $$x$$ and $$y$$, is it true that $$x

• Transitivity : for all $$x,y,z$$, is it true that $$x

Can you finish? If you have a question don't hesitate.

• Symmetry: for all $x$ and $\mathbf y$ ... – J. W. Tanner May 3 at 19:37