Question

Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, given justification for this
(i) On $Z^{+}$, define $\ast$ by $a\ast b=a-b$
(ii) On $Z^{+}$, define $\ast$ by $a\ast b=ab$
(iii) On $R$, define $\ast$ by $a\ast b=a{b}^2$
(iv) On $Z^{+}$, define $\ast$ by $a\ast b=|a-b|$
On $Z^{+}$, define $\ast$ by $a\ast b=a$

Answer 1

(i) On $Z^{+}$, the binary operation $\ast$ defined by $a\ast b=a-b$ is not a binary operation because if the points are taken as $\left(1,2\right)$, then by applying binary operation, it becomes $1-2=-1$ and $-1$ does not belong to $Z^{+}$.

(ii) On $Z^{+}$, the binary operation $\ast$ defined by $a\ast b=ab$ is a binary operation because each element in $Z^{+}$ has a unique element in $Z^{+}$.

(iii) On $R$, the binary operation $\ast$ defined by $a\ast b=a{b}^2$ is a binary operation because each element in $R$ has a unique element in$R$.

(iv) On $Z^{+}$, the binary operation $\ast$ defined by $a\ast b=\left|a-b\right|$ is a binary operation because each element in $Z^{+}$ has a unique element in $Z^{+}$.

(v) On $Z^{+}$, the binary operation $\ast$ defined by $a\ast b=a$ is a binary operation because each element in $Z^{+}$ has a unique element in $Z^{+}$.