Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation R = {((a1, b1), (a

Q: Wait, why do we define a new set $S$? Can't we just list all pairs in $A \times B$?

That is a fantastic question! While you could brute-force list all elements of $A \times B$, which has $3 \times 3 = 9$ elements, the relation $R$ is defined on $(A \times B) \times (A \times B)$, which would have $9 \times 9 = 81$ possible pairs. Listing all 81 and checking the condition for each is a recipe for disaster and silly mistakes. By defining $S$, we isolate the 'divisibility' logic, turning a complex problem into a simple counting exercise. It is about working smarter, not harder!

Q: Why are we multiplying the counts? Why not add them?

Think about the Fundamental Principle of Counting. We have two independent conditions: $a_1 | b_2$ and $a_2 | b_1$. For every valid pair $(a_1, b_2)$ that satisfies the first condition, we can pair it with any valid pair $(a_2, b_1)$ that satisfies the second condition. If you have 6 choices for the first and 6 for the second, the total combinations are $6 \times 6$. Addition would only be appropriate if the conditions were mutually exclusive, which they are not!

Q: Does the order of $(a_1, b_1)$ and $(a_2, b_2)$ matter?

Absolutely! The relation $R$ is a set of ordered pairs of ordered pairs. The element $((a_1, b_1), (a_2, b_2))$ is distinct from $((a_2, b_2), (a_1, b_1))$ unless the pairs themselves are identical. The structure is rigid, and the conditions are specific to the positions of $a_1, b_2, a_2,$ and $b_1$. Always respect the structure defined in the problem statement!

Question

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

Options:

(A) 36

(B) 24

(C) 18

(D) 12

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a complex structure. You have two sets, $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$.

At first glance, this looks like a simple set theory problem. However, the relation $R$ is defined on the Cartesian product $(A 	imes B) 	imes (A 	imes B)$.

This is a relation between ordered pairs, which often causes confusion. Let us deconstruct this step-by-step.

### The Power of Decoupling

The definition of $R$ is given as:

$$R = \{((a_1, b_1), (a_2, b_2)) \in (A 	imes B) 	imes (A 	imes B) : a_1 | b_2 	ext{ and } a_2 | b_1\}$$

The key to unlocking this is to recognize that the conditions $a_1 | b_2$ and $a_2 | b_1$ are independent.

We can define an auxiliary set $S = \{(a, b) \in A 	imes B : a | b\}$. By doing this, we transform the problem into finding the number of elements in $S$ and then squaring that count.

### Building the Set $S$

Let us systematically find the elements of $S$ by iterating through each element of $A$ and checking which elements of $B$ it divides.

For $a = 2$, we check $B = \{8, 9, 12\}$. Since $2$ divides $8$ and $12$, we have the pairs $(2, 8)$ and $(2, 12)$.

For $a = 3$, we check $B$. Since $3$ divides $9$ and $12$, we have the pairs $(3, 9)$ and $(3, 12)$.

For $a = 4$, we check $B$. Since $4$ divides $8$ and $12$, we have the pairs $(4, 8)$ and $(4, 12)$.

Gathering these, we find:

$$S = \{(2, 8), (2, 12), (3, 9), (3, 12), (4, 8), (4, 12)\}$$

The cardinality of $S$, denoted as $n(S)$, is exactly $6$.

### The Final Synthesis

Now, we return to the relation $R$. The condition $a_1 | b_2$ means the pair $(a_1, b_2)$ must be an element of $S$.

Since there are $6$ such pairs in $S$, there are $6$ choices for $(a_1, b_2)$. Similarly, the condition $a_2 | b_1$ means the pair $(a_2, b_1)$ must be an element of $S$, giving us another $6$ choices.

Because these choices are independent, the Fundamental Principle of Counting tells us to multiply them:

$$n(R) = n(S) 	imes n(S) = 6 	imes 6 = 36$$

The total number of elements in the relation $R$ is **36**. You have successfully navigated the complexity of relations by breaking them down into their fundamental components.

Animated Solution for Mathematics - Sets and Relations: Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$ . Then the number of elements in the relation $R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1} divides b_{2} and a_{2} divides b_{1}}$ is

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Visualized Solution

Understanding the Given Sets $A$ and $B$

Analyzing the Structure of Relation $R$

Deconstructing the Conditions of $R$

Defining the Divisibility Set $S$

Finding Elements of $S$ : Case $a = 2$

Finding Elements of $S$ : Case $a = 3$

Finding Elements of $S$ : Case $a = 4$

Total Elements in Set $S$

Applying the Fundamental Principle of Counting

Applying the Second Condition

Final Calculation of $n (R)$

The Sigma Insight: Cartesian Product of Sets

Analyzing the Setup

The Power of Decoupling

Building the Set $S$

The Final Synthesis

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Visualized Solution

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Analyzing the Structure of Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Deconstructing the Conditions of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

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