Question

If $ A = \{a, b, c\}, B = \{b, c, d\} $ and $ C = \{a, d, c\} $, then $ (A - B) \times (B \cap C) $ is equal to

• A. \( \{(a, c), (a, d)\} \)
• B. \( \{(a, b), (c, d)\} \)
• C. \( \{(c, a), (d, a)\} \)
• D. \( \{(a, c), (a, d), (b, d)\} \)

Hint:

Remember that the Cartesian product involves pairing every element from the first set with every element from the second set. For difference and intersection operations, carefully evaluate the resulting sets before forming the product.

Answer 1

The Correct Option is A

To solve the problem $ (A - B) \times (B \cap C) $, we proceed as follows:

Step 1: Find the set difference $A - B$. This is defined as the set of elements that are in $A$ but not in $B$. For sets A and B given by:
$ A = \{a, b, c\} $
$ B = \{b, c, d\} $
We have:
$ A - B = \{a\} $ (since 'a' is only in A).

Step 2: Find the intersection $B \cap C$. The intersection of two sets is the set of elements that are common to both sets. For sets B and C given by:
$ B = \{b, c, d\} $
$ C = \{a, d, c\} $
We have:
$ B \cap C = \{c, d\} $ (since 'c' and 'd' are common).

Step 3: Compute the Cartesian product $(A - B) \times (B \cap C)$. The Cartesian product of two sets is the set of all ordered pairs, where the first element is from the first set and the second is from the second set. Thus:
$(A - B) \times (B \cap C) = \{(a, c), (a, d)\}$.

Therefore, the solution is:

\(\{(a, c), (a, d)\}\)

Answer 2

Let's first solve for \( A - B \) and \( B \cap C \): \[ A - B = \{a, b, c\} - \{b, c, d\} = \{a\} \] \[ B \cap C = \{b, c, d\} \cap \{a, d, c\} = \{c, d\} \] Now, the Cartesian product of \( (A - B) \) and \( (B \cap C) \) is: \[ (A - B) \times (B \cap C) = \{a\} \times \{c, d\} = \{(a, c), (a, d)\} \] Thus, the answer is \( \{(a, c), (a, d)\} \).