Question

Which of the following statements is false for sets A, B and C?

• A. \(A \cap (B - C) = (A \cup B) - (A \cap B)\)
• B. \(A - (B - C) = (A - B) \cup (A \cap C)\)
• C. \(A - B = A \cap B^c\)
• D. \(A \subseteq B\) iff \(A \cup B = B\)

Hint:

Exam Tip:
For set theory identity questions:

• Draw Venn diagrams to verify.
• Use algebraic laws (De Morgan's laws, distributive laws).
• Test with small sets if unsure.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the properties of set operations. We need to identify which of the given statements is false.

Step 2: Analyzing Each Option:
Let's check each option using set theory laws.
• Option (A): \(A \cap (B - C) = (A \cup B) - (A \cap B)\)
Left Hand Side (LHS): \(A \cap (B \cap C^c) = A \cap B \cap C^c\).
Right Hand Side (RHS): \((A \cup B) - (A \cap B) = (A \cup B) \cap (A \cap B)^c\).
This is the symmetric difference of A and B, which is \((A - B) \cup (B - A)\).
These two are not equal in general.
For example, let \(A = \{1, 2\}\), \(B = \{2, 3\}\), \(C = \{3\}\).
LHS: \(A \cap (B - C) = \{1, 2\} \cap (\{2, 3\} - \{3\}) = \{1, 2\} \cap \{2\} = \{2\}\).
RHS: \((A \cup B) - (A \cap B) = \{1, 2, 3\} - \{2\} = \{1, 3\}\).
LHS \(\neq\) RHS. So, option (A) is false.
• Option (B): \(A - (B - C) = (A - B) \cup (A \cap C)\)
LHS: \(A - (B \cap C^c) = A \cap (B \cap C^c)^c = A \cap (B^c \cup C) = (A \cap B^c) \cup (A \cap C) = (A - B) \cup (A \cap C)\).
This matches the RHS. So, option (B) is true.
• Option (C): \(A - B = A \cap B^c\)
By definition, \(A - B = \{x \mid x \in A, x \notin B\} = A \cap B^c\).
This is true.
• Option (D): \(A \subseteq B\) iff \(A \cup B = B\)
If \(A \subseteq B\), then \(A \cup B = B\). Conversely, if \(A \cup B = B\), then \(A \subseteq B\).
This is a standard set theory result and is true.

Step 3: Final Answer:
Only option (A) is false. Therefore, option (A) is correct.