Question

The difference A - B between two sets A and B is equal to

• A. $A \cup B^c$
• B. $A \cap B^c$
• C. $A^c \cap B$
• D. $A^c \cup B^c$

Hint:

Venn diagrams are useful for visualizing set differences. Shading the area of $A$ that does not overlap with $B$ reveals exactly the intersection of $A$ and the exterior of $B$ ($B^c$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The set difference $A - B$ represents the set of all elements that are members of set $A$ but are not members of set $B$.
Key Formula or Approach:
Using standard set theory identities, the relative complement can be written in terms of intersection and absolute complement:
\[ A - B = \{ x \mid x \in A \text{ and } x \notin B \} \]

Step 2: Detailed Explanation:
Let us analyze the logical definition of $A - B$:
- The condition $x \in A$ corresponds to set $A$.
- The condition $x \notin B$ corresponds to the complement of set $B$, denoted as $B^c$ (or $B'$).
- The word "and" represents the set intersection ($\cap$).
Combining these parts, we get:
\[ A - B = A \cap B^c \]
Therefore, the difference $A - B$ is equivalent to $A \cap B^c$.

Step 3: Final Answer
The correct option is (B).