R S Aggarwal Solutions for Exercise 2: EXERCISE

Consider a binary operation on $Q–\left\{1\right\}$, defined by $a * b = a + b-ab$. Find the identity element in $Q–\left\{1\right\}$.

 Let $Q_0$ be the set of all nonzero rational numbers. Let $*$be a binary operation on $Q_0$, defined by $a^{*}b=\frac{ab}{4}$ for all $a, b \in Q_0$. Find the identity element $Q_0$.

 Let $Q_0$ be the set of all nonzero rational numbers. Let $*$be a binary operation on $Q_0$, defined by $a^{*}b=\frac{ab}{4}$ for all $a, b \in Q_0$. Find the inverse of an element $a$ in $Q_0$.

Let $A=N\times N$. Define $*$ on $A$ by $(a,b)*(c,a)=(a+c,b+d)$. Show that identity element does not exist in $A$.

Let $A=(1,-1, i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that multiplication is associative on $A$.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that multiplication is commutative on $A$.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that $1$ is the multiplicative identity.

Let $A=(1,-1\mathit{,}i,-i)$ be the set of four $4^{th}$ roots of unity. Prepare the composition table for multiplication on $A$ and show that every element in $A$ has its multiplicative inverse.