$(2$ points$)$ Determine if the following functions are

one$-$to$-$one, onto, both, or neither.

a$.f$:$RZ,f(x)={|}_{{?}_{?}}2x-1_{{?}_{?}}|$

b$.f$:$RR,f(x)=\{\begin{array}{ccc}2x+4,& if& x<mn\; id="EditorElement\_1871251">3</mn>\\ x+7,& if& x3\end{array}$

$(4$ points$)$ For each of the following relations,

determine if they are reflexive, symmetric,

antisymmetric, and$/$or transitive. Also indicate if

the relation is an equivalence relation or not.

a$.\{(m,n)||m-n|1\}$

on the set of

integers

b$.\{(m,n)|m^2=n^2\}$

on the set of

integers

c$.$

$\{a,b,c\}$

d$.\{(x,x),(y,z),(z,y)\}$

on the set

$\{x,y,z\}$

$(2$ points$)f(x)=3x+1$ and $g(x)=4x-2,$ are

functions from $R$ to $R.$ Find

a$.f$@$g$

b$.g$@$f$

$(2$ points$)$ Compute the sum ${\displaystyle \underset{m=3}{\overset{n}{}}}(4m+5).$