UGC NET CSE | June 2012 | Part 3 | Question: 17

Question

Let $Q(x,y)$ denote “x+y=0” and let there be two quantifications given as

1. $\exists y \forall x Q(x,y)$
2. $\forall x \exists y Q(x,y)$

where $x$ and $y$ are real numbers. Then which of the following is valid?

• A. I is true and II is false
• B. I is false and II is true
• C. I is false and II is also false
• D. both I and II are true

Comments

answer for this question should be both false ,i think question is not correct

Answer 1

if then for all x there exists some y   x+y=0  (e.g  take any number as x then some y =-x will always be there)

in symbolic form it is written as  ∀x ∃y Q(x,y) so second  is true

  

now for some y all x are not here to hold x+y=0  for some y some x are there so first  is not true

hence Ans is B

Comments

Both are same.. If first statement is true, then 2nd should also be true... Why 2nd is wrong accoriding to u

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The actual question is 

Answer 2

Option (B) is correct i.e. (i) is false & (ii) is true. 

Explanation. The universe of discourse is the set of all things we wish to talk about that is, the set of all objects that we can sensibly assign to a variable in a propositional function. Let us consider that ∃y∀x Q(x, y) There exists a y for every x equation x+y=0. Let x=1 then y should be -1. Let x=2 then y should be -2. So it is different for every value of X. So it is for every x there exists a y where x+y=0. So premises 2 is right.