Determine if the following functions are one$-$to$-$one, onto, both, or neither.

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

b$.f$:$RR,f(x)=\{\begin{array}{ccc}2x+4,& if& x<mn\; id="EditorElement\_956691">3</mn>\\ x+7,& if& x3\end{array}$

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$.\{(a,a),(b,b),(c,c),(a,b),(a,c),(c,b)\}$ on the set $\{a,b,c\}$

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