If * be binary operation defined on $R$ by $a * b=1+a b, \forall a, b \in R$. Then, the operation $*$ is
(i) commutative but not associative.
(ii) associative but not commutative.
(iii) neither commutative nor associative.
(iv) both commutative and associative.
$$\begin{aligned} &\text { (i) Given that, }\\ &\begin{aligned} & a * b=1+a b, \forall a, b \in R \\ & a * b=a b+1=b * a \end{aligned} \end{aligned}$$
So, * is a commutative binary operation.
Also,
$$\begin{aligned} a *(b * c) & =a *(1+b c)=1+a(1+b c) \\ a *(b * c) & =1+a+a b c \quad\text{.... (i)}\\ (a * b) * c & =(1+a b) * c \\ & =1+(1+a b) c=1+c+a b c\quad\text{.... (ii)} \end{aligned}$$
So, * is not associative.
Hence, * is commutative but not associative.
Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$ be defined as $a R b$, if $a$ is congruent to $b, \forall a, b \in T$. Then, $R$ is
Consider the non-empty set consisting of children in a family and a relation $R$ defined as $a R b$, if $a$ is brother of $b$. Then, $R$ is
The maximum number of equivalence relations on the set $A=\{1,2,3\}$ are
If a relation $R$ on the set $\{1,2,3\}$ be defined by $R=\{(1,2)\}$, then $R$ is