GATE CS 2019 Set 1 — Question 15
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Engineering Mathematics → Sets & Combinatorics → Permutations & Combinations
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Question
Let . Let . Consider the following two statements on .
I.
II.
Which of the above statements is/are TRUE?
I.
II.
Which of the above statements is/are TRUE?
Correct answer
(C) Both I and II
Solution
We want to find the size of the set .Method 1 (Counting by element ):
For a fixed element (there are choices for ), we need to choose a subset such that . This is equivalent to choosing any subset of . The number of such subsets is .
Total count .
Thus, Statement I is TRUE.Method 2 (Counting by subset size ):
Let . The number of subsets of size is . For each such subset, there are choices for (since ).
Summing over all possible sizes from 1 to :
.
Thus, Statement II is TRUE.Since both statements are true, the correct option is (C).
For a fixed element (there are choices for ), we need to choose a subset such that . This is equivalent to choosing any subset of . The number of such subsets is .
Total count .
Thus, Statement I is TRUE.Method 2 (Counting by subset size ):
Let . The number of subsets of size is . For each such subset, there are choices for (since ).
Summing over all possible sizes from 1 to :
.
Thus, Statement II is TRUE.Since both statements are true, the correct option is (C).
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