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which of the following options are equivalent to statement?

 

in Mathematical Logic by (64 points) | 23 views
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option A
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correct, how did you do it...

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$\forall xP(x) \vee \exists yP(y)$ is true

$\implies$ either $\forall xP(x)$ is true or  $\exists yP(y)$ is true …..$(i)$

 

Now, if $P(x)$ is true for all $x$ then $P(x)$ will be true for some $x$ also right ?

i.e. $\exists xP(x)$ is true , put this in eq $(i)$

$\implies$ either $\exists xP(x)$ is true or  $\exists yP(y)$ is true

$\implies \exists xP(x) \vee \exists yP(y)$ is true

write $x$ in place of $y$

$\implies \exists xP(x) \vee \exists xP(x)$ is true

$\implies \exists xP(x) $ is true

$\therefore$ Option A is correct.

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