UGCNET-Dec2012-III: 75

Question

Let $\theta(x, y, z)$ be the statement “x+y=z” and let there be two quantification given as

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

where x, y, z are real numbers, then which one of the following is correct?

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

Answer 1

x+y=z  for all x for all y there exist some Z which will satisfy this equation as e.g x=4351 y=1111 then some Z =5462 is there and so on

for some z say z=100 there do not exist all x and all y (there exist only some x, y ) which satisfies this equation hence II is false

 so ans is B

Comments

@Sir Sanjay Sharma,Sir can you explain your example a bit more ?

Answer 2

D) I is false and II is false

I) it means for each and every x and y there exists  z. which is False according to the equation x+y=z

II) it means there exists a z for all x and all y , which is also False.

Because here for each x and each y there exists a single z