Determine whether or not the definition of * given below gives a … - Vidyadip

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 $Z^+$ define * by $a * b = |a - b|$
Here, $Z^+$ denotes the set of all non-negative integers.

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Answer

On $Z^+, *$ is defined by $a * b = |a - b|.$
It is seen that for each $\text{a, b}\in\text{Z}^{+},$ there is a unique element $|a - b|$ in $Z^+.$
This means that $*$ carries each pair $(a, b)$ to a unique element $a * b = |a - b|$ in $Z^+.$
Therefore, $*$ is a binary operation.

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