Let A = {1, 2, 3, 4, 5, 6, 7}. Then the relation $R=\left\{\right(x,y)\in A\times A:x+y=7\}$ is                     [2023]

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation

$R=\left\{\right((a_1,b_1),(a_2,b_2))\in (A\times B,A\times B):a_1$ divides $b_2$ and $a_2$ divides $b_1\}$ is                   [2023]

Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10}. Let R be a relation defined on A x B such that $R=\left\{\right((a_1,b_1),(a_2,b_2)):a_1\le b_2$ and $b_1\le a_2\}.$ Then the number of elements in the set R is             [2023]

Let R be a relation on R, given by

R = $\left\{\right(a,b):3a-3b+\sqrt{7}$ is an irrational number$\}.$

Then R is                                                         [2023]

Let P(S) denote the power set of S = {1, 2, 3, ...., 10}. Define the relations $R_1$ and $R_2$ on P(S) as $A{R}_{1}B$ if $(A\cap B^C)\cup (B\cap A^C)=\varphi$ and $A{R}_{2}B$ if $A\cup B^C=B\cup A^C,\forall A,B\in P\left(S\right)$. Then                  [2023]

The relation $R=\left\{\right(a,b):gcd(a,b)=1,2a\ne b,a,b,\in Z\}$ is                      [2023]

Let R be a relation defined on $\mathrm{\mathbb{N}}$ as $a R b$ if $2a+3b$ is a multiple of 5, $a,b\in \mathrm{\mathbb{N}}$.                   [2023]

The minimum number of elements that must be added to the relation R = {(a, b), (b, c)} on the set {a, b, c} so that it becomes symmetric and transitive is              [2023]

Let R be a relation on N x N defined by (a, b) R(c, d) if and only if $ad(b-c)=bc(a-d)$. Then R is                  [2023]

Among the relations

$S=\left\{\right(a,b):a,b\in \mathit{R}-\{0\},2+\frac{a}{b}>0\}$ and 

$T=\left\{\right(a,b):a,b\in \mathit{R}\mathbf{,}{a}^{\mathbf{2}}-b^2\in Z\}$                                      [2023]