Determine whether or not each of the definition of given below gives a binary operation.

Question:

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) On $\mathbf{Z}^{+}$, define ${ }^{*}$ by $a{ }^{*} b=a-b$

(ii) $\mathrm{On} \mathbf{Z}^{+}$, define ${ }^{*}$ by $a^{*} b=a b$

(iii) On $\mathbf{R}$, define * ${ }^{*}$ by $a^{*} b=a b^{2}$

(iv) On $\mathbf{Z}^{+}$, define ${ }^{*}$ by $a^{*} b=|a-b|$

(v) On $\mathbf{Z}^{+}$, define ${ }^{*}$ by $a^{*} b=a$

Solution:

(i) On $\mathbf{Z}^{+}$, ${ }^{*}$ is defined by $a^{*} b=a-b$.

It is not a binary operation as the image of $(1,2)$ under ${ }^{*}$ is $1^{*} 2=1-2$ $=-1 \notin Z^{+}$.

(ii) $\mathrm{On} \mathbf{Z}^{+}{ }^{*}{ }^{*}$ is defined by $a{ }^{*} b=a b$.

It is seen that for each $a, b \in \mathbf{Z}^{+}$, there is a unique element $a b$ in $\mathbf{Z}^{+}$.

This means that * carries each pair $(a, b)$ to a unique element $a^{*} b=a b$ in $\mathbf{Z}^{+}$.

Therefore, ${ }^{*}$ is a binary operation.

(iii) On $\mathbf{R}$, ${ }^{*}$ is defined by $a^{*} b=a b^{2}$.

It is seen that for each $a, b \in \mathbf{R}$, there is a unique element $a b^{2}$ in $\mathbf{R}$.

This means that * carries each pair $(a, b)$ to a unique element $a$ * $b=a b^{2}$ in $\mathbf{R}$.

Therefore, ${ }^{*}$ is a binary operation.

(iv) On $\mathbf{Z}^{+},{ }^{*}$ is defined by $a^{*} b=|a-b|$.

It is seen that for each $a, b \in \mathbf{Z}^{+}$, there is a unique element $|a-b|$ in $\mathbf{Z}^{+}$.

This means that * carries each pair $(a, b)$ to a unique element $a{ }^{*} b=$ $|a-b|$ in $\mathbf{Z}^{+}$.

Therefore, ${ }^{*}$ is a binary operation.

(v) $\mathrm{On} \mathbf{Z}^{+}{ }^{*}{ }^{*}$ is defined by $a^{*} b=a$.

${ }^{*}$ carries each pair $(a, b)$ to a unique element $a^{*} b=a$ in $\mathbf{Z}^{+}$.

Therefore, ${ }^{*}$ is a binary operation.