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Question
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On Z+, defined * by a * b = a − b
Here, Z+ denotes the set of all non-negative integers.
Solution
If a = 1 and b = 2 in Z+, then
\[a * b = a - b\]
\[ = 1 - 2\]
\[ = - 1 \not\in Z^+ \left[ \because Z^+ \text{ is the set of non-negative integers } \right]\]
\[ \Rightarrow \text{Fora} = 1 \text{ and }b = 2, \]
\[a * b \not\in Z^+ \]
\[\text{Thus}, * \text{ is not a binary operation on } Z^+ .\]
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