Question

If \( A = \{1,2\}, B = \{1,2,3,4\}, C = \{5,6\}, D = \{5,6,7,8\} \), then state which of the following is true?

• A. \( (A \times C) \subset (B \times D) \)
• B. \( A \times B \subset C \times D \)
• C. \( (A \times B) \subset (A \times D) \)
• D. \( A \times C \subset B \times D \)

Hint:

Cartesian product follows subset rule: \( A\subset B, C\subset D \Rightarrow A\times C \subset B\times D \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Cartesian product \( P \times Q \) is a set of all ordered pairs \( (p, q) \) such that \( p \in P \) and \( q \in Q \).
For \( (P \times Q) \subset (X \times Y) \) to be true, it is necessary that \( P \subset X \) and \( Q \subset Y \).
Step 2: Detailed Explanation:
Let's check the subsets for option A:
Check if \( A \subset B \): A = \{1, 2\}, B = \{1, 2, 3, 4\}. All elements of A are in B, so \( A \subset B \).
Check if \( C \subset D \): C = \{5, 6\}, D = \{5, 6, 7, 8\}. All elements of C are in D, so \( C \subset D \).
Since \( A \subset B \) and \( C \subset D \), any pair \( (a, c) \) where \( a \in A \) and \( c \in C \) must also satisfy \( a \in B \) and \( c \in D \).
Therefore, \( (A \times C) \subset (B \times D) \).
Step 3: Final Answer:
The correct statement is (A \(\times\) C) \(\subset\) (B \(\times\) D).