The shaded portion in the Venn diagram of set (B - A) is: Which of the following is correct? (a) Both Assertion and Reason are true and Reason is the correct explanation of Asserti... The shaded portion in the Venn diagram of set (B - A) is: Which of the following is correct? (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion. (b) Both Assertion and Reason are true but Reason is not the correct explanation of Assertion. (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true.
Understand the Problem
The question involves analyzing a Venn diagram related to set theory and determining the correct interpretation of an assertion and reasoning about combinations. Specifically, it asks for the identification of the shaded area in a Venn diagram of the set (B - A) and evaluating the truth of assertions regarding combinations of selections.
Answer
The shaded portion in the Venn diagram of set \( B - A \) is option (d).
Answer for screen readers
The shaded portion in the Venn diagram of set ( B - A ) is option (d).
Steps to Solve
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Understanding Set Difference The expression ( B - A ) represents all elements in set ( B ) that are not in set ( A ). The shaded area in the Venn diagram for ( B - A ) would typically depict only the part of ( B ) that does not overlap with ( A ).
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Identifying the Shaded Area in the Venn Diagram Look at each option (a, b, c, d) in the Venn diagram. You're looking for the area that includes only elements in ( B ) and excludes any elements that also belong to ( A ).
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Evaluating the Assertion and Reason The assertion states that the number of ways to select 8 items from 20 is the same as selecting 12 items. This can be evaluated using the combinations formula: $$ C(n, r) = \frac{n!}{r!(n-r)!} $$ Check if ( C(20, 8) ) is equal to ( C(20, 12) ).
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Analyzing the Combinations Using the property of combinations: $$ C(n, r) = C(n, n-r) $$ Here, ( C(20, 8) = C(20, 20-8) ) and ( C(20, 12) = C(20, 20-12) ). Thus, ( C(20, 8) = C(20, 12) ).
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Determining the Correctness of Assertion and Reason Since both the assertion and reason are true and the reason provides an explanation for the assertion, choose the correct option accordingly.
The shaded portion in the Venn diagram of set ( B - A ) is option (d).
More Information
In this context, the assertion regarding combinations is significant because it illustrates how the properties of combinatorial selection can often simplify what might appear to be complex relationships.
Tips
- Misunderstanding what ( B - A ) represents in the Venn diagram. It specifically indicates elements unique to ( B ), not those shared with ( A .
- Not applying the property of combinations correctly. Remembering that ( C(n, r) = C(n, n-r) ) can help check assertions effectively.
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