R D Sharma Solutions for Chapter: Binary Operations, Exercise 1: EXERCISE

Let $A=R_0\times R,$ where $R_0$ denote the set of all non-zero real numbers. A binary operation $\mathbf{\text{'}}O\mathbf{\text{'}}\mathbf{ }$is defined on $A$ as follows: $(a,b)O(c,d) = (ac,bc+d)$for all $(a,b), (c,d) \in R_0\times R$. Show that $\text{'}O\text{'}$ is not commutative but associative on $A$.

Let $A=R_0\times R,$ where $R_0$ denote the set of all non-zero real numbers.A binary operation $\mathbf{\text{'}}O\mathbf{\text{'}}\mathbf{ }$is defined on $A$ as follows $(a,b)O(c,d) = (ac,bc+d)$for all $(a,b), (c,d) \in R_0\times R$. Find the identity element in $A$.

Let $\text{'}o\text{'}$ be a binary operation on the set $Q_0$ of all non-zero rational numbers defined by $aob=\frac{ab}{2},$ for all $a,b \in Q_0$. Find the identity element in $Q_0.$

Let$\text{'}o\text{'}$ be a binary operation on the set $Q_0$ of all non-zero rational numbers defined by $aob=\frac{ab}{2}$, for all $a,b\in Q_0$. Find the invertible elements of $Q_0$.

Let$R_0$ denote the set of all non-zero real numbers and let $A=R_0\times R_0$. If $\text{'}*\text{'}$ is a binary operation on$A$ defined by
$(a, b) * (c, d) = (ac, bd)$ for all $(a, b), (c, d) \in A$. Show that $\text{'}*\text{'}$is both commutative and associative on$A ?$

Let $R_O$ denote the set of all non-zero real numbers and let $A=R_O\times R_{O_{}}.$ If '*' is a binary operation on $A$ defined by
$(a, b) * (c, d) = (ac, bd)$ for all $(a, b), (c, d) \in A$
Find the identity element in $A ?$

Let $*$ be the binary operation on $N$ defined by$a*b= HCF$ of $a$ and $b$.
Does there exist identity for this binary operation on $N$

Let $A = R\times R$ and $*$ be a binary operation onAdefined by $(a,b)*(c,d)=(a+c,b+d).$ Show that$*$ is commutative and associative. Find the binary element for $*$ on $A$, if any.