Sure! Let's break down each part of the problem step by step.
### Part (a)
We are given the Cartesian product \(\( P \times Q = \{(4, 5), (4, 6), (4, 7), (5, 5), (5, 6), (5, 7)\} \)\).
#### Step 1: Find P
We can notice that in the ordered pairs of \(\( P \times Q \)\), the first component is either 4 or 5. Therefore, set \(\( P \)\) must be:
\(\[ P = \{4, 5\} \]\)
#### Step 2: Find Q
Similarly, from the ordered pairs, the second component is 5, 6, or 7. Therefore, set \(\( Q \)\) must be:
\(\[ Q = \{5, 6, 7\} \]\)
#### Step 3: Calculate \(\( n(P \times Q) \)\)
The number of elements in the Cartesian product \(\( P \times Q \)\) can be calculated as follows:
\(\[ n(P \times Q) = \text{Number of elements in } P \times \text{Number of elements in } Q \]\)
Given that:
\(\[ n(P) = 2 \]\)
and
\(\[ n(Q) = 3 \]\)
Therefore:
\(\[ n(P \times Q) = 2 \times 3 = 6 \]\)
#### Step 4: Verification
We have:
\(\[ n(P \times Q) = 6 \]\)
\(\[ n(P) \times n(Q) = 2 \times 3 = 6 \]\)
Both calculations confirm that:
\(\[ n(P \times Q) = n(P) \times n(Q) \]\)
### Part (b)
We are given the Cartesian product \(\( R \times S = \{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)\} \)\).
#### Step 1: Find R
From the ordered pairs of \(\( R \times S \)\), the first component is either 1 or 2. Therefore, set \(\( R \)\) must be:
\(\[ R = \{1, 2\} \]\)
#### Step 2: Find S
Similarly, from the ordered pairs, the second component is 2, 3, or 4. Therefore, set \(\( S \)\) must be:
\(\[ S = \{2, 3, 4\} \]\)
#### Step 3: Calculate \(\( n(R \times S) \)\)
The number of elements in the Cartesian product \(\( R \times S \)\) can be calculated as follows:
\(\[ n(R \times S) = \text{Number of elements in } R \times \text{Number of elements in } S \]\)
Given that:
\(\[ n(R) = 2 \]\)
and
\(\[ n(S) = 3 \]\)
Therefore:
\(\[ n(R \times S) = 2 \times 3 = 6 \]\)
#### Step 4: Verification
We have:
\(\[ n(R \times S) = 6 \]\)
\(\[ n(R) \times n(S) = 2 \times 3 = 6 \]\)
Both calculations confirm that:
\(\[ n(R \times S) = n(R) \times n(S) \]\)
So, we have found the sets \(\( P, Q, R, \)\) and \(\( S \)\), and we have verified that the number of elements in the Cartesian products \(\( P \times Q \)\) and \(\( R \times S \)\) is equal to the product of the number of elements in the respective sets.
