Class 12 Mathematics Relation and Function Binary Operation

Medium

Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.

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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.

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Relation and Function

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Binary Operation

On the set of positive rationals,a binary operation $$ is defined by $a b = \frac{2ab}{5}$. If $2 * x = 3^{-1}$,then $x = $

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Determine whether or not each of the definitions of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^+$,define $$ by $a b = |a - b|$.

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Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a b = (a - b)^2$. Determine whether the operation is commutative and associative.

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Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

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Determine which of the following binary operations on the set $N$ are associative and which are commutative. $a * b = \frac{a+b}{2}$ for all $a, b \in N$.

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Show that addition and multiplication are associative binary operations on $R$. However,subtraction is not associative on $R$,and division is not associative on $R_*$.

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