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. (i) On Z+, defined * by a * b = a – b (ii) On Z+, define * by a*b = ab
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. (i) On Z+, defined * by a * b = a – b (ii) On Z+, define * by a*b = ab

(i)Since, On Z+, defined * by a * b = a – b

If a = 1 and b = 2 in Z+, then

    \[\begin{array}{*{35}{l}} a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }-\text{ }b  \\ =\text{ }1\text{ }-\text{ }2  \\ =\text{ }-1\text{ }\notin \text{ }{{Z}^{+~}}\left[ because\text{ }{{Z}^{+}}~is\text{ }the\text{ }set\text{ }of\text{ }non-negative\text{ }integers \right]  \\ \end{array}\]

For a = 1 and b = 2,

a * b ∉ Z+

Thus, * is not a binary operation on Z+.

(ii) Given Z+, define * by a*b = a b

Let

    \[\begin{array}{*{35}{l}} a,\text{ }b\text{ }\in \text{ }{{Z}^{+}}  \\ \Rightarrow \text{ }a,\text{ }b\text{ }\in \text{ }{{Z}^{+}}  \\ \Rightarrow \text{ }a\text{ }*\text{ }b\text{ }\in \text{ }{{Z}^{+}}  \\ \end{array}\]

Thus, * is a binary operation on R.