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 :
Assertion (A) Correctness: Set Identity $(A \cup B) - A$
The assertion states that for non-empty sets A and B, the identity $(A \cup B) - A = A \cap B^C$ is true. Let's examine this.
Evaluating $(A \cup B) - A$
- Using the definition of set difference, $X - Y = X \cap Y^C$: $(A \cup B) - A = (A \cup B) \cap A^C$
- Applying the distributive law of set intersection over union: $(A \cup B) \cap A^C = (A \cap A^C) \cup (B \cap A^C)$
- The intersection of a set with its complement is the empty set ($A \cap A^C = \emptyset$): $(A \cap A^C) \cup (B \cap A^C) = \emptyset \cup (B \cap A^C)$
- Simplifying the union with the empty set: $\emptyset \cup (B \cap A^C) = B \cap A^C$
- The expression $B \cap A^C$ is the definition of the set difference $B - A$.
Thus, $(A \cup B) - A$ simplifies to $B - A$. The assertion claims it equals $A \cap B^C$, which is equivalent to $A - B$. Since $B - A$ is not generally equal to $A - B$, Assertion (A) is incorrect.
Reason (R) Validity: Venn Diagrams
Reason (R) states that Venn diagrams help to prove some set relations. Venn diagrams are visual tools that effectively represent sets and their relationships through overlapping circles. They are commonly used to understand and demonstrate set operations like union, intersection, and complement, and are indeed helpful in verifying set identities.
Therefore, Reason (R) is correct.
Final Conclusion
Based on the step-by-step analysis, Assertion (A) is incorrect, and Reason (R) is correct. This aligns with the conclusion that (A) is not correct but (R) is correct.
Similar Questions
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then arrange the following in non-decreasing order.
A. $|A \cup B|$
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C. $|A \cup B \cup C|$
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Choose the correct answer from the options given below : Which of the following are true for the given chart ?
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B. 20,000 is the difference between the total production of C and B in the given years.
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Which of the following are true on the basis of data given in chart ?
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C. Cat, Dog, Elephant III. 
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- What is the key guide to statecraft in the realist tradition?
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A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
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B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
