Let A = {1, 4, 7} and B = {2, 3, 8}. Then the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $S = A 	imes B = \{(1,2), (1,3), (1,8), (4,2), (4,3), (4,8), (7,2), (7,3), (7,8)\}$.
The relation $R$ is defined on $S 	imes S$, where $|S|

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(D) Let $S = A 	imes B = \{(1,2), (1,3), (1,8), (4,2), (4,3), (4,8), (7,2), (7,3), (7,8)\}$.
The relation $R$ is defined on $S 	imes S$, where $|S| = 9$, so $|S 	imes S| = 81$.
We need to find pairs $((a_1, b_1), (a_2, b_2))$ such that $(a_1 + a_2)$ divides $(b_1 + b_2)$.
Let $x_1 = (a_1, b_1)$ and $x_2 = (a_2, b_2)$.
We test combinations of $a_1, a_2 \in \{1, 4, 7\}$ and $b_1, b_2 \in \{2, 3, 8\}$:
$1$. If $a_1+a_2=2$ (i.e.,$a_1=1, a_2=1$), then $b_1+b_2$ must be even. Possible $(b_1, b_2)$ are $(2,2), (2,8), (3,3), (8,2), (8,8)$. ($5$ pairs)
$2$. If $a_1+a_2=8$ (i.e.,$(1,7), (7,1), (4,4)$), then $b_1+b_2$ must be a multiple of $8$.
- For $(1,7)$, $b_1+b_2$ can be $8$ or $16$. Pairs: $(2,8), (8,2), (8,8)$. ($3$ pairs)
- For $(7,1)$, $b_1+b_2$ can be $8$ or $16$. Pairs: $(2,8), (8,2), (8,8)$. ($3$ pairs)
- For $(4,4)$, $b_1+b_2$ can be $8$ or $16$. Pairs: $(2,8), (8,2), (8,8)$. ($3$ pairs)
$3$. If $a_1+a_2=14$ (i.e.,$(7,7)$), then $b_1+b_2$ must be a multiple of $14$. No pair sums to $14$. ($0$ pairs)
$4$. If $a_1+a_2=5$ (i.e.,$(1,4), (4,1)$), then $b_1+b_2$ must be a multiple of $5$. Pairs: $(2,3), (3,2)$. ($2$ pairs each, total $4$ pairs)
$5$. If $a_1+a_2=11$ (i.e.,$(4,7), (7,4)$), then $b_1+b_2$ must be a multiple of $11$. No pair sums to $11$. ($0$ pairs)
Summing these up: $5 + 3 + 3 + 3 + 4 = 18$. Re-evaluating the condition $a_1+a_2$ divides $b_1+b_2$ for all $81$ pairs, the total count is $18$.

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + a_2 \text{ divides } b_2 + b_1\}$ is . . . . . . .

Similar Questions

$f(x) = \frac{x}{\ln x}$ and $g(x) = \frac{\ln x}{x}$. Identify the $CORRECT$ statement.

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) = \text{the sum of positive divisors of } n$, then $f(2^k \cdot 3)$, where $k$ is a positive integer, is

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$,$x \in R$,then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

Let $f:[0,1] \rightarrow \mathbb{R}$ be an injective continuous function that satisfies the condition $-1 < f(0) < f(1) < 1$. Then,the number of functions $g:[-1,1] \rightarrow [0,1]$ such that $(g \circ f)(x) = x$ for all $x \in [0,1]$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module