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Is the given statement True ?  Please explain

• $\forall_x \left \{ P(x) \vee Q(x) \right \}\Leftrightarrow \forall_x P(x) \vee \forall_x Q(x)$
• $\forall_x \left \{ P(x) \wedge Q(x) \right \}\Leftrightarrow \forall_x P(x) \wedge \forall_x Q(x)$
• $\exists_x \left \{ P(x) \vee Q(x)\right \} \Leftrightarrow \exists_x P(x) \vee \forall_x Q(x)$
• $\exists_x \left \{ P(x) \wedge Q(x) \right \}\Leftrightarrow \exists_x P(x) \wedge \exists_x Q(x)$

Only 2nd and third is true
∀ is true when it is true for each nd every value of its domain so it is distributive on AND
∃ is true when it is true for atleast value of its domain  so it is distributive on OR
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Either try with examples or you need to prove it(which is difficult)

Its hard for me try with examples also :(
Is there any other approach ?

Can u just show for this http://gateoverflow.in/3783/gate2005-it-36

How to check other options using examples ? :(
Sorry i was asking too much questions . I was not able to understand this topic at all

They have given example for 2005 question. Try it on your own. You will get it
Even I find difficult taking examples.

+1 vote