Chapter 1.7, Problem 2E

In each of the following parts, a relation $R$ is defined on the set $Z$ 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 $Z$.

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 $xy\ge 0$.

i. $xRy$ if and only if $xy\le 0$.

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

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