Determine whether or not each of the definitions of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^+$,define $*$ by $a * b = |a - b|$.
(B) On $Z^+$,the operation $*$ is defined by $a * b = |a - b|$.
For any two elements $a, b \in Z^+$,the result $|a - b|$ must be an element of $Z^+$ for $*$ to be a binary operation.
Consider $a = 1$ and $b = 1$. Both $1, 1 \in Z^+$.
Then $a * b = |1 - 1| = 0$.
Since $0 \notin Z^+$ (as $Z^+$ is the set of positive integers ${1, 2, 3, ...}$),the operation $*$ does not map every pair $(a, b)$ to an element in $Z^+$.
Therefore,$*$ is not a binary operation on $Z^+$.
For any two elements $a, b \in Z^+$,the result $|a - b|$ must be an element of $Z^+$ for $*$ to be a binary operation.
Consider $a = 1$ and $b = 1$. Both $1, 1 \in Z^+$.
Then $a * b = |1 - 1| = 0$.
Since $0 \notin Z^+$ (as $Z^+$ is the set of positive integers ${1, 2, 3, ...}$),the operation $*$ does not map every pair $(a, b)$ to an element in $Z^+$.
Therefore,$*$ is not a binary operation on $Z^+$.
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