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Question

# Which of the following is true? (a) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Z (b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q (c) all binary commutative operations are associative (d) subtraction is a binary operation on N

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Solution

## (b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q. Let us check each option one by one. (a) $\text{If}a=1\mathrm{and}b=2,\phantom{\rule{0ex}{0ex}}a*b=\frac{a+b}{2}\phantom{\rule{0ex}{0ex}}=\frac{1+2}{2}\phantom{\rule{0ex}{0ex}}=\frac{3}{2}\notin Z$ Hence, (a) is false. (b) $a*b=\frac{a+b}{2}\in Q,\forall a,b\in Q\phantom{\rule{0ex}{0ex}}\mathrm{For}\mathrm{example}:\mathrm{Let}a=\frac{3}{2},b=\frac{5}{6}\in Q\phantom{\rule{0ex}{0ex}}a*b=\frac{\frac{3}{2}+\frac{5}{6}}{2}\phantom{\rule{0ex}{0ex}}=\frac{9+5}{12}\phantom{\rule{0ex}{0ex}}=\frac{14}{12}\phantom{\rule{0ex}{0ex}}=\frac{7}{6}\in Q$ Hence, (b) is true. (c) Commutativity: $\text{Let}a,b\in N.\mathrm{Then},\phantom{\rule{0ex}{0ex}}a*b={2}^{ab}\phantom{\rule{0ex}{0ex}}={2}^{ba}\phantom{\rule{0ex}{0ex}}=b*a\phantom{\rule{0ex}{0ex}}\text{Therefore,}\phantom{\rule{0ex}{0ex}}a*b=b*a,\forall a,b\in N$ Thus, * is commutative on N. Associativity: $\text{Let}a,b,c\in N.\mathrm{Then},\phantom{\rule{0ex}{0ex}}a*\left(b*c\right)=a*\left({2}^{bc}\right)\phantom{\rule{0ex}{0ex}}={2}^{a*{2}^{bc}}\phantom{\rule{0ex}{0ex}}\left(a*b\right)*c=\left({2}^{ab}\right)*c\phantom{\rule{0ex}{0ex}}={2}^{ab*{2}^{c}}\phantom{\rule{0ex}{0ex}}\text{Therefore,}\phantom{\rule{0ex}{0ex}}a*\left(b*c\right)\ne \left(a*b\right)*c\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ Thus, * is not associative on N. Therefore, all binary commutative operations are not associative. Hence, (c) is false. (d) Subtraction is not a binary operation on N because subtraction of any two natural numbers is not always a natural number. For example: 2 and 4 are natural numbers. 2$-$4 = $-$2 which is not a natural number. Hence, (d) is false.

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