Determine whether or not each of the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation, give justification for this.

(i) Or Z+, define $*$ by a$*$ b$=$ a$-$b

(ii) On Z+, define $*$ by a$*$ b $=$ ab

(iii) On R, define $*$ by a $*$ b $= ab^2$

(iv) On Z+, define $*$ by a $*$ b $= |a - b|$

(v) On Z+, define $*$ by a $*$ b $=$ a

(i) Z+ $= \{1, 2, 3,...\}$, we have a $*$ b $=$ a$-$b

Let a $=$ 1, b $=$ 3 $\Rightarrow$ a$*$ 6 $=$ 1$-$3 $=$ $-$2 $\notin$Z+

Hence, the operation $*$ is not a binary operation on Z+.

(ii) Z+$= \{1, 2, 3, \}$, we have a $*$ b $=$ ab

Let a $=$ 2, 6 $=$ 4 $\Rightarrow$ a$*$b $=$ 2 $*$ 4 $=$ 8 $\in$Z+

Hence, the operation $*$ is a binary operation on Z+.

(iii) R (set of real numbers), we have a $*$ b $= ab^2$

Let a $=$ 5.2, b $=$ 3 $\Rightarrow$ a$*$b $=$ 5.2(3)2 $=$ 46.8 $\in$R

Hence, the operation $*$ is a binary operation on R.

(iv) Z+ $= \{1, 2, 3,....\}$, we have a $*$ b $= |a - b|$

Let a $=$3, b $=$ 7 $\Rightarrow$ a $*$ b $=$ |3$-$7| $= | - 7| = 4 \in {Z^ + }$

Hence, the operation $*$ is a binary operation on Z+.

(v) Z+ $= \{1, 2, 3,....\}$, we have a $*$ b $=$ a

Let a$=$5, b$=$7 $\Rightarrow$ a$*$b $=$ 5 $\in {Z^ + }$

Hence, the operation $*$ is a binary operation on ${Z^ + }.$