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GATE | GATE-CS-2001 | Question 27

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  • Last Updated : 28 Jun, 2021
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Consider the following statements:

S1: There exists infinite sets A, B, C such that 
    A ∩ (B ∪ C) is finite.
S2: There exists two irrational numbers x and y such
    that (x+y) is rational.

Which of the following is true about S1 and S2?
(A) Only S1 is correct
(B) Only S2 is correct
(C) Both S1 and S2 are correct
(D) None of S1 and S2 is correct


Answer: (C)

Explanation: S1: A ∩ (B ∪ C)
Here S1 is finite where A, B, C are infinite
We’ll prove this by taking an example.
Let A = {Set of all even numbers} = {2, 4, 6, 8, 10…}
Let B = {Set of all odd numbers} = {1, 3, 5, 7………..}
Let C = {Set of all prime numbers} = {2, 3, 5, 7, 11, 13……}
B U C = {1, 2, 3, 5, 7, 9, 11, 13……}
A ∩ (B ∪ C)
Will
be equals to: {2} which is finite.
I.e. using A, B, C as infinite sets the statement S1 is finite.
So, statement S1 is correct.
S2: There exists two irrational numbers x, y such that (x+y) is rational
To prove this statement as correct, we take an example.
Let X = 2-Sqrt (3), Y = 2+Sqrt (3) => X, Y are irrational
X+Y = 2+Sqrt (3) + 2-Sqrt (3) = 2+2 = 4
So, statement S2 is also correct.
Answer is Option C

Both Statements S1, S2 are correct.

 

This solution is contributed by Anil Saikrishna Devarasetty.

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