Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.
For any $a, b, c \in R$,addition is associative because $(a + b) + c = a + (b + c)$.
Multiplication is associative because $(a \times b) \times c = a \times (b \times c)$.
Subtraction is not associative because $(8 - 5) - 3 = 3 - 3 = 0$,while $8 - (5 - 3) = 8 - 2 = 6$. Since $0 \neq 6$,subtraction is not associative.
Division is not associative on $R_*$ (where $R_* = R \setminus \{0\}$) because $(8 \div 5) \div 3 = \frac{8}{5} \div 3 = \frac{8}{15}$,while $8 \div (5 \div 3) = 8 \div \frac{5}{3} = \frac{24}{5}$. Since $\frac{8}{15} \neq \frac{24}{5}$,division is not associative.
Multiplication is associative because $(a \times b) \times c = a \times (b \times c)$.
Subtraction is not associative because $(8 - 5) - 3 = 3 - 3 = 0$,while $8 - (5 - 3) = 8 - 2 = 6$. Since $0 \neq 6$,subtraction is not associative.
Division is not associative on $R_*$ (where $R_* = R \setminus \{0\}$) because $(8 \div 5) \div 3 = \frac{8}{5} \div 3 = \frac{8}{15}$,while $8 \div (5 \div 3) = 8 \div \frac{5}{3} = \frac{24}{5}$. Since $\frac{8}{15} \neq \frac{24}{5}$,division is not associative.
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Similar Questions
Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^* \,3) \,^* \,(4 \,^* \,5)$.
(Hint: use the following table)
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
(Hint: use the following table)
| $^*$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $2$ | $1$ | $2$ | $2$ | $2$ | $2$ |
| $3$ | $1$ | $2$ | $3$ | $3$ | $3$ |
| $4$ | $1$ | $2$ | $3$ | $4$ | $4$ |
| $5$ | $1$ | $2$ | $3$ | $4$ | $5$ |
Medium
View SolutionIf $a * b = 10ab$ on $Q^{+}$,then find the inverse of $0.01$.
Medium
View SolutionShow that the operation $*: R \times R \rightarrow R$ defined by $a * b = a + 2b$ is not associative.
Medium
View SolutionShow that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.
Medium
View SolutionLet $*^{\prime}$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$. Is the operation $*^{\prime}$ same as the operation $*$ defined in Exercise $4$ above? Justify your answer.
Medium
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