GATE CS 2014 Set 2 — Question 60
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Engineering Mathematics → Sets & Combinatorics → Partial Orders & Lattices
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Question
Consider the following relation on subsets of the set of integers between 1 and 2014. For two distinct subsets and of we say if the minimum element in the symmetric difference of the two sets is in .Consider the following two statements:
: There is a subset of that is larger than every other subset.
: There is a subset of that is smaller than every other subset.Which one of the following is CORRECT?
: There is a subset of that is larger than every other subset.
: There is a subset of that is smaller than every other subset.Which one of the following is CORRECT?
Correct answer
(A) Both S1 and S2 are true
Solution
The relation defined is a strict total order on the power set of . Let denote the symmetric difference . The condition is if .Analyzing Statement S1 (Largest Element):
Consider the empty set . For any non-empty set , the symmetric difference . The minimum element is , which is clearly in . Therefore, by the definition, for all . Thus, is the largest element. is true.Analyzing Statement S2 (Smallest Element):
Consider the set itself. For any proper subset , the symmetric difference . The minimum element of is an element of . Therefore, , which implies for all . Thus, is the smallest element. is true.Since both statements are true, option (A) is correct.
Consider the empty set . For any non-empty set , the symmetric difference . The minimum element is , which is clearly in . Therefore, by the definition, for all . Thus, is the largest element. is true.Analyzing Statement S2 (Smallest Element):
Consider the set itself. For any proper subset , the symmetric difference . The minimum element of is an element of . Therefore, , which implies for all . Thus, is the smallest element. is true.Since both statements are true, option (A) is correct.
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