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Consider the first order predicate formula $\varphi$:

$\forall x [ ( \forall z \: z | x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z | w \Rightarrow ((w=z) \vee (z=1)))]$

Here $a \mid b$ denotes that  $ ’a \text { divides } b’ $ , where  $a$ and $b$ are integers. Consider the following sets:

  • $S_1: \{1, 2, 3, \dots , 100\}$
  • $S_2:$ Set of all positive integers
  • $S_3:$ Set of all integers

Which of the above sets satisfy $\varphi$?

  1. $S_1$ and $S_2$
  2. $S_1$ and $S_3$
  3. $S_2$ and $S_3$
  4. $S_1, S_2$ and $S_3$
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10 Answers

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$\text{The definition of z and w is clearly saying that both w and x are prime numbers as they are divisible by themselves or 1.}$

∀x { ∃w | w<x }

$\text{w,x are Prime numbers}$

$\text{This means for each element in set if that element is prime number, you will definately see its brother who is also prime but he is bigger than him.}$

 

Hence  A1 is false as for 97, next big prime is out of the range.

A2 is true.

Also A3 is true as It reduces to A2 only in terms of solving

 

 

 

Answer:

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