Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = (a - b)^{2}$ for all $a, b \in Q$ is commutative.
- ACommutative
- BNot commutative
- CAssociative
- DNone of these
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Similar Questions
Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.
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View SolutionFor each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Z^+$,define $a ^* b = a^b$.
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View SolutionConsider a binary operation $*$ on the set $\{1,2,3,4,5\}$ given by the following multiplication table. Is $^*$ commutative?
(Hint: use the following table)
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
(Hint: use the following table)
| $^*$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $2$ | $1$ | $2$ | $2$ | $2$ | $2$ |
| $3$ | $1$ | $2$ | $3$ | $3$ | $3$ |
| $4$ | $1$ | $2$ | $3$ | $4$ | $4$ |
| $5$ | $1$ | $2$ | $3$ | $4$ | $5$ |
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View SolutionOn the set of all natural numbers $N$,which one of the following $*$ is a binary operation?
Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.
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