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$($ $∀x A(x)$  $∪$   $∀x B(x)$ $)$  $→$  $∀x$  $[$ $A(x)$  $ ∪ $  $B(x)] $

$Is$  $this$ $formula$  $VALID$?
in Mathematical Logic by (308 points) | 17 views
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yes, it is VALID.

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$(\forall x A(x) \vee \forall x B (x) )\rightarrow \forall x [A(x) \vee B(x)]$

If we found a case which can lead to $T \rightarrow F$ then we can say that it is not valid.

$(\forall x A(x) \vee \forall x B (x) )$ will be true if all x are true for A or all x are true for B or all x are true for Both A and B.

To make $\forall x [A(x) \vee B(x)]$ false we need x which is false for both A and B but this is contradiction because LHS says that all x are true for A or all x are true for B or all x are true for Both A and B.

Hence the given wff is Valid.

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