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# 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, give justification for this - On$\; Z^+,\,$ defined $*\,$ by$\; a*b= a-b$

Note: This is part 1 of a 5 part question, split as 5 separate questions here.

Toolbox:
• A binary operation $∗$ on a set $A$ is a function $∗$ from $A \times A$ to $A$. Therefore, if $a,b \in A \Rightarrow a*b \in A\; \forall\; a,b, \in A$
Given On $Z^+ *$ is defined by $a*b=a-b$
Consider $a=1, b = 2$ under $* \Rightarrow 1 * 2 = 1- 2 = -1$.
However, $-1 \notin Z^+$, therefore $*$ is not a binary operation.
edited Mar 19, 2013