Textbook Problem

A relation $R$ on a nonempty set $A$ is called irreflexive if $x\cancel{R}x$ for all $x\in A$. Which of the relations in Exercise 2 are irreflexive?

In each of the following parts, a relation $R$ is defined on the set $\mathbb{Z}$ of all integers. Determine in each case whether or not $R$ is reflexive, symmetric, or transitive. Justify your answers.

a. $xRy$ if and only if $x=2y.$

b. $xRy$ if and only if $x=-y.$

c. $xRy$ if and only if $y=xk$ for some $k$ in $\mathbb{Z}$.

d. $xRy$ if and only if $x<y.$

e. $xRy$ if and only if $x\ge y.$

f. $xRy$ if and only if $x=\left|y\right|.$

g. $xRy$ if and only if $\left|x\right|\le \left|y+1\right|.$

h. $xRy$ if and only if $xy\ge \mathrm{}$

i. $xRy$ if and only if $xy\le \mathrm{}$

j. $xRy$ if and only if $\left|x-y\right|=1$.

k. $xRy$ if and only if $\left|x-y\right|<1$.