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 ${Z^ + }.$
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