Odisha Board Solutions for Chapter: Relation and Function, Exercise 3: Exercise- 1(c)

Show that the operation $*$ given by $x*y=x+y-xy$ is a binary operation on $Z,Q$ and $R$ but not on $N$.

Determine whether the following operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

$a*b=$LCM $\left\{a,b\right\}$  on $\left\{0,1,2,3,4,..........,10\right\}$.

Determine whether the following operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

$a*b=\sqrt{a^2+b^2}$ on $Q$.

Determine whether the following operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

$a*b=a\times b$(mod$5$) on $\left\{0,1,2,3,4\right\}$.

Determine whether the following operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

$a*b=a+b-ab$ on $R-\left\{1\right\}$.

In case $*$ is a binary operation in $a*b=2a+3b$ on $Z$, test whether it is $\left(i\right)$ associative $\left(ii\right)$ commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.

In case $*$ is a binary operation in $a*b=ma-nb$ on $Q^{+}$ where $m$ and $n\in N$, test whether it is $\left(i\right)$ associative $\left(ii\right)$ commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.

Construct the composition table/multiplication table for the binary operation $*$ defined on $\left\{0,1,2,3,4\right\}$ by $a*b=a\times b$(mod $5$). Find the identity element if any. Also find the inverse elements of $2$ and $4$.