For each of the following relations on the set of all integers, determine whether the relation is reflexive, symmetric, and/or transitive:
a. (,)∈ if and only if <.
b. (,)∈ if and only ≥1.
c. (,)∈ if and only =−.
d. (,)∈ if and only =||
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.
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 $$ x<y \implies y<x \text{ (a). }\\ xy \ge 1 \implies yx \ge 1 \text{ (b). }\\ x = -y \implies y = -x \text{ (c). }\\ x = |y| \implies y = |x| \text{ (d). } $$
Transitivity : for all $x,y,z$, is it true that $$ x<y \text{ and } y<z \implies x<z \text{ (a). }\\ xy \ge 1 \text{ and } yz \ge 1 \implies xz \ge 1 \text{ (b). }\\ x = -y \text{ and } y = -z \implies x = -z \text{ (c). }\\ x = |y| \text{ and } y = |z| \implies x = |z| \text{ (d). } $$
Can you finish? If you have a question don't hesitate.