© All Rights reserved @ LearnWithDash
Step-by-Step Detailed Solution
Step 1: Identify the Sets and Elements
We have two sets:
$A = \{2,\,3,\,4\}$
$B = \{8,\,9,\,12\}$.
The Cartesian product $A \times B$ consists of ordered pairs $(a,\,b)$ where $a \in A$ and $b \in B$.
Step 2: Understand the Relation $R$
The relation $R$ is defined on $A \times B$. So each element of $R$ is of the form:
$\bigl((a_1,\,b_1),\,(a_2,\,b_2)\bigr)$,
with the conditions:
1. $a_1$ divides $b_2$,
2. $a_2$ divides $b_1$.
Step 3: Find All Valid Pairs for "a divides b"
We first list all valid pairs $(a,\,b)$ with $a \in A$ and $b \in B$ for which $a$ divides $b$:
$2$ divides $8$ → $(2,\,8)$
$2$ divides $12$ → $(2,\,12)$
$3$ divides $9$ → $(3,\,9)$
$3$ divides $12$ → $(3,\,12)$
$4$ divides $8$ → $(4,\,8)$
$4$ divides $12$ → $(4,\,12)$
So there are $6$ valid pairs in $A \times B$ that satisfy "$a$ divides $b$".
Step 4: Apply Conditions to Form Elements of $R$
An element of $R$ is $\bigl((a_1,\,b_1),\, (a_2,\,b_2)\bigr)$ such that:
• $a_1$ divides $b_2$, and
• $a_2$ divides $b_1$.
We have $6$ possible pairs whenever we need "$a$ divides $b$". The key observation is that these choices are independent:
– For the first condition $(a_1,\,b_2)$, we can pick any of the $6$ valid pairs from our list.
– For the second condition $(a_2,\,b_1)$, we can again pick any of the $6$ valid pairs.
Therefore, for each choice of $(a_1,\,b_1)$ matching one divisibility condition, there are $6$ ways to choose $(a_2,\,b_2)$ matching the other condition, giving a total of $6 \times 6$ combinations.
Step 5: Calculate the Number of Elements
Hence, the number of elements in the relation $R$ is
$6 \times 6 = 36$.
Final Answer
The number of elements in the relation $R$ is $\boxed{36}$.
