Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If A and B are non-empty sets then (A \(\cup\) B) - A = A \(\cap\) B\(^{C}\) is true.
Reason (R): Venn diagrams help to prove the some set relations.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

If you are unsure of the algebraic laws of set theory during an exam, quickly sketch a 2-circle Venn diagram.
Shade \((A \cup B) - A\) (which is only the B region) and \(A \cap B^C\) (which is only the A region).
You will immediately see they do not overlap, proving Assertion (A) is false!

Answer 1

The Correct Option is D

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Step 1: Understanding the Question: This question is based on set theory operations and identities.
We need to test the algebraic identity presented in Assertion (A) and the statement in Reason (R).

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Step 2: Key Formula or Approach:
We can simplify set expressions using set laws:
1. Set difference law: \(X - Y = X \cap Y^C\)
2. Distributive and De Morgan's laws.
Let us simplify LHS and RHS of Assertion (A) to check if they are identical.

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Step 3: Detailed Explanation:
- Evaluating Assertion (A):
LHS:
\[ (A \cup B) - A \]
Using the set difference law:
\[ (A \cup B) - A = (A \cup B) \cap A^C \]
Applying the distributive law:
\[ (A \cup B) \cap A^C = (A \cap A^C) \cup (B \cap A^C) \]
Since \(A \cap A^C = \emptyset\):
\[ \emptyset \cup (B \cap A^C) = B \cap A^C = B - A \]
So, LHS represents elements that are only in B (excluding those in A).
RHS:
\[ A \cap B^C = A - B \]
RHS represents elements that are only in A (excluding those in B).
Since \(B - A \neq A - B\) for general non-empty sets \(A\) and \(B\) (unless \(A = B\)), the assertion statement is incorrect.
Thus, Assertion (A) is not correct.
- Evaluating Reason (R):
Venn diagrams are visual representations of sets and are widely used to verify or prove fundamental relationships between sets.
Therefore, the statement "Venn diagrams help to prove some set relations" is correct.
Since Assertion (A) is false and Reason (R) is true, we conclude that option (D) is the correct answer.

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Step 4: Final Answer: (A) is not correct but (R) is correct, matching option (D).