Algebra of Relations

The binary operation $*R\times R\to R,$ is defined as  $a*b=2a+b.$ From the given options choose the value of  $\left(2*3\right)*4$ is equal to

Among the two statements

$\left(S_1\right):\left(p\Rightarrow q\right)\wedge \left(p\wedge \left(~q\right)\right)$ is a contradiction and $\left(S_2\right):\left(p\wedge q\right)\vee \left(\left(~p\right)\wedge q\right)\vee \left(p\wedge \left(~q\right)\right)\vee \left(\left(~p\right)\wedge \left(~q\right)\right)$ is a tautology

Let $A=\left\{2,3,4\right\}$ and $B=\left\{8,9,12\right\}$. Then the number of elements in the relation  $R=\left\{\left(\left(a_1,b_1\right),\left(a_2,b_2\right)\right)\in \left(A\times B,A\times B\right):a_1 \text{divides} b_2 \text{and} a_2 \text{divides} b_1\right\}$ is

Let $A=\left\{0,3,4,6,7,8,9,10\right\}$ and $R$ be the relation defined on $A$ such that $R\left\{\left(x,y\right)\in A\times A:x-y \text{is odd positive integer} \text{or} x-y=2\right\}$. The minimum number of elements that must be added to the relation $R,$ so that it is a symmetric relation, is equal to $\_\_\_\_\_\_\_\_\_$

If $R$ is a relation on finite set $A$ having $n$ elements, then the number of relations on $A$ is

Let $A$ and $B$ be two smallest sets such that $A\cup \left\{1\right\}=\left\{1,2,3,4\right\}$ and $B\cup \left\{5\right\}=\left\{4,5,6,7,8\right\}.$ If $P=A-B$ and $Q=B-A$, then the number of relations from $P$ to $Q$ is

let $*$ be a binary operation on $R$defined by $a*b=\frac{a}{4}+\frac{b}{7}$,$a,b\in R$.Show that $*$ is not commutative and associative.

Show that division is not a binary operation on real numbers$\left(R\right)$. Give one example.

Show that subtraction and division are not binary operations on $N$.

$A=\left\{3,9\right\}$, $B=\left\{9,6\right\}$ and $C=\left\{3,6\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,4\right\}$, $B=\left\{4,5\right\}$ and $C=\left\{1,5\right\}$. Verify associative property of Cartesian product of sets.

$A=\left\{1,2\right\}$, $B=\left\{2,3\right\}$ and $C=\left\{1,3\right\}$. Verify associative property of Cartesian product of sets.

If $A=\left\{5,9\right\}$, $B=\left\{6,8\right\}$ and $C=\left\{9,7\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{b,a\right\}$, $B=\left\{c,e\right\}$ and $C=\left\{a,d\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

If $A=\left\{0,1\right\}$, $B=\left\{2,4\right\}$ and $C=\left\{1,3\right\}$, then prove that, $A\times (B\cup C)=(A\times B)\cup (A\times C)$.

Suppose $*$ be the multiplication operation and $\#$ be the addition operation defined on $Z$. Let $a=10,b=11 \text{and} c=12$, then find $a*(b\#c)$.

Suppose $*$ be the multiplication operation and $\#$ be the addition operation defined on $Z$. Let $a=5,b=6 \text{and} c=8$, then find $a*(b\#c)$.

Suppose $*$ be the multiplication operation and $\#$ be the addition operation defined on $Z$. Let $a=3,b=4 \text{and} c=7$, then find $a*(b\#c)$.