Q.9 Given the following sets:
𝐴 = {2, 4, 6, 8, 10, 12}
𝐵 = {8, 10, 12, 14, 16, 18}
𝐶 = {7, 8, 9, 10 11, 12, 13}
(𝐴 ∩ 𝐵) ∪ (𝐵 ∩ 𝐶) is
(A) {8, 10, 12, 14}
(B) {8, 10, 12}
(C) {7, 8, 10, 11, 12, 13, 14}
(D) {4, 6, 7, 8 10, 11, 12, 13}
(A ∩ B) ∪ (B ∩ C) equals {8, 10, 12}, matching option (B). This result comes from finding the common elements between A and B, then B and C, and combining those without duplicates. Detailed steps and option analysis follow below.
Step-by-Step Solution
First, compute A ∩ B, the elements common to both A = {2, 4, 6, 8, 10, 12} and B = {8, 10, 12, 14, 16, 18}. The intersection yields {8, 10, 12}. Next, compute B ∩ C, where C = {7, 8, 9, 10, 11, 12, 13}; the common elements are also {8, 10, 12}. Finally, union these: {8, 10, 12} ∪ {8, 10, 12} = {8, 10, 12}.
Option Analysis
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(A) {8, 10, 12, 14}: Includes 14 from B but not in A ∩ B or B ∩ C, so incorrect.
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(B) {8, 10, 12}: Matches exact result of both intersections being identical.
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(C) {7, 8, 10, 11, 12, 13, 14}: Adds C-unique elements like 7, 11, 13 and 14, expanding beyond intersections.
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(D) {4, 6, 7, 8, 10, 11, 12, 13}: Mixes A-unique (4,6), C-unique (7,11,13), missing key logic.
Mastering set theory operations like (A ∩ B) ∪ (B ∩ C) builds essential skills for CSIR NET Life Sciences quantitative reasoning, especially in genetics and data interpretation. This guide breaks down the exact problem with sets A = {2, 4, 6, 8, 10, 12}, B = {8, 10, 12, 14, 16, 18}, and C = {7, 8, 9, 10, 11, 12, 13}, revealing why option (B) {8, 10, 12} is correct.
Keyphrase Focus
Target (A ∩ B) ∪ (B ∩ C) appears naturally in queries for intersection union problems, CSIR NET set theory MCQs, and practical set operation examples.
Why This Matters for Exams
Set intersections identify shared elements, while unions combine without repetition—core to analyzing gene overlaps or ecological datasets in life sciences. Practice reveals patterns: here, B ∩ C ⊆ A ∩ B, simplifying to {8, 10, 12}.
Common Pitfalls
Students often confuse with A ∪ B or full unions, adding extras like 14 or 7, as in wrong options. Venn diagrams visualize: shade A-B overlap, B-C overlap, then union both.
