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, given justification for this(i) On ${Z^ + }$ , define $*$ by $a * b = a - b$ (ii) On ${Z^ + }$ , define $*$ by $a * b = ab$ (iii) On $R$ , define $*$ by $a * b = a{b^2}$ (iv) On ${Z^ + }$ , define $*$ by $a * b = \left| {a - b} \right|$ (v) On ${Z^ + }$ , define $*$ by $a * b = a$

(i) On ${Z^ + }$ , the binary operation$*$ defined by $a * b = a - b$ 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$*$ defined by$a * b = ab$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(iii) On $R$ , the binary operation$*$ defined by $a * b = a{b^2}$ is a binary operation because each element in $R$ has a unique element in $R$ .
(iv) On ${Z^ + }$ , the binary operation $*$ defined by $a * b = \left| {a - b} \right|$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .
(v) On ${Z^ + }$ , the binary operation $*$ defined by $a * b = a$ is a binary operation because each element in ${Z^ + }$ has a unique element in ${Z^ + }$ .