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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?

  1. I is true and II is true
  2. I is true and II is false
  3. I is false and II is true
  4. I is false and II is false
in Discrete Mathematics
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2 Answers

4 votes
 
Best answer
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

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

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

Answer:

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