Question

Suppose that the number of elements in set A is p, the number of elements in set B is q and the number of elements in \(A \times B\) is 7 then \(p^2 + q^2 =\)

• A. 50
• B. 42
• C. 51
• D. 49

Answer 1

The Correct Option is A

The number of elements in the Cartesian product \(A \times B\) is given by the product of the number of elements in set \(A\) and the number of elements in set \(B\). So, we have:

\[ |\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| = 7 \]

Since \( |\vec{A} \times \vec{B}| = 7 \), we know that \( p \cdot q = 7 \).

To find the value of \( p^2 + q^2 \), we need to find the possible values of \(p\) and \(q\) that satisfy \( p \cdot q = 7 \).

The possible pairs of \((p, q)\) that satisfy \( p \cdot q = 7 \) are \((1, 7)\) and \((7, 1)\).

For both pairs, we have:

\[ p^2 + q^2 = 1^2 + 7^2 = 1 + 49 = 50 \]

Therefore, \( p^2 + q^2 = 50 \).

The correct option is (A) 50.

Answer 2

Given: Number of elements in \( A \times B = 7 \)
We know that:
\[ n(A \times B) = p \cdot q = 7 \] Since 7 is a prime number, possible values of \( p \) and \( q \) are:

• \( p = 1, q = 7 \Rightarrow p^2 + q^2 = 1^2 + 7^2 = 1 + 49 = 50 \)
• \( p = 7, q = 1 \Rightarrow p^2 + q^2 = 7^2 + 1^2 = 49 + 1 = 50 \)

Correct Answer: 50

Answer 3

Let \(n(A)\) denote the number of elements in set A, and \(n(B)\) denote the number of elements in set B. 

We are given:

• \(n(A) = p\)
• \(n(B) = q\)
• \(n(A \times B) = 7\)

The number of elements in the Cartesian product of two sets A and B is given by the formula:

\[ \mathbf{n(A \times B) = n(A) \times n(B)} \]

Substituting the given values into the formula:

\[ 7 = p \times q \]

Since \(p\) and \(q\) represent the number of elements in sets, they must be positive integers (assuming the sets are non-empty, which they must be if their product has 7 elements).

The number 7 is a prime number. The only positive integer factors of 7 are 1 and 7.

Therefore, the only possible pairs of positive integers \((p, q)\) such that \(p \times q = 7\) are:

• Case 1: \(p = 1\) and \(q = 7\)
• Case 2: \(p = 7\) and \(q = 1\)

We need to find the value of \(p^2 + q^2\).

In Case 1 (\(p = 1, q = 7\)):

\[ p^2 + q^2 = (1)^2 + (7)^2 = 1 + 49 = 50 \]

In Case 2 (\(p = 7, q = 1\)):

\[ p^2 + q^2 = (7)^2 + (1)^2 = 49 + 1 = 50 \]

In both possible cases, the value of \(p^2 + q^2\) is 50.

Comparing this with the given options, the correct option is:

50