# Help me justify if these statements are logically equivalent

• Sep 6th 2009, 08:46 AM
zpwnchen
Help me justify if these statements are logically equivalent
1) Determine if $\displaystyle \forall x(P(x) \rightarrow Q(x))$ and $\displaystyle \forall xP(x) \rightarrow \forall xQ(x)$ are logically equivalent. Justify your answers.

2) Determine if $\displaystyle \forall x(P(x) \leftrightarrow Q(x))$ and $\displaystyle \forall xP(x) \leftrightarrow \forall xQ(x)$ are logically equivalent. Justify your answers.

3) Show that $\displaystyle \exists x(P(x) \vee Q(x))$ and $\displaystyle \exists xP(x) \vee \exists xQ(x)$ are logically equivalent.

I'm getting confused with these. Please explain me in a simpliest way. Thank you very much for your help.
• Sep 6th 2009, 10:29 AM
Plato
Quote:

Originally Posted by zpwnchen
1) Determine if $\displaystyle \forall x(P(x) \rightarrow Q(x))$ and $\displaystyle \forall xP(x) \rightarrow \forall xQ(x)$ are logically equivalent. Justify your answers.

2) Determine if $\displaystyle \forall x(P(x) \leftrightarrow Q(x))$ and $\displaystyle \forall xP(x) \leftrightarrow \forall xQ(x)$ are logically equivalent. Justify your answers.

3) Show that $\displaystyle \exists x(P(x) \vee Q(x))$ and $\displaystyle \exists xP(x) \vee \exists xQ(x)$ are logically equivalent.

Here are some suggestions.

1) $\displaystyle \forall x(P(x) \rightarrow Q(x))$ means "every P is a Q".
$\displaystyle \forall xP(x) \rightarrow \forall xQ(x)$ means "if everything is a P then everything is a Q"
Do those mean the same thing?

3) $\displaystyle \exists x(P(x) \vee Q(x))$ means "something is a P or it is a Q"
$\displaystyle \exists xP(x) \vee \exists xQ(x)$ means something is a P or something is a Q"
Do those mean the same thing?
• Sep 6th 2009, 01:00 PM
zpwnchen
1) $\displaystyle \forall x(P(x) \rightarrow Q(x))$ means "every P is a Q".
$\displaystyle \forall xP(x) \rightarrow \forall xQ(x)$ means "if everything is a P then everything is a Q"
Do those mean the same thing?
I feel that it's not the same thing.

3) $\displaystyle \exists x(P(x) \vee Q(x))$ means "something is a P or it is a Q"
$\displaystyle \exists xP(x) \vee \exists xQ(x)$ means something is a P or something is a Q"
Do those mean the same thing?

I do not get this one :(

Is it possible to create truth table for these?
How about the second one? Are they logically equivalent?
• Sep 6th 2009, 01:08 PM
Plato
What if I said "Something is red or blue."
Then said "Something is red or something is blue"
Are those saying exactly the same thing?
• Sep 6th 2009, 02:18 PM
zpwnchen
Quote:

Originally Posted by Plato
What if I said "Something is red or blue."
Then said "Something is red or something is blue"
Are those saying exactly the same thing?

I suppose they have the same meaning. Right?

And for the second problem, they are not logically equivalent,right?

Can we create truth table for that?
• Sep 6th 2009, 02:53 PM
Plato
Quote:

Originally Posted by zpwnchen
And for the second problem, they are not logically equivalent,right? CORRECT
Can we create truth table for that?

The proofs depend on the rules for instantiation and generalization.
I have never seen truth-tables used in this connection.
Here are the standard statements of these three.
$\displaystyle \left( {\forall x} \right)\left[ {P(x) \Rightarrow Q(x)} \right] \Rightarrow \left( {\left( {\forall x} \right)\left[ {P(x)} \right] \Rightarrow \left( {\forall x} \right)\left[ {Q(x)} \right]} \right)$
$\displaystyle \left( {\forall x} \right)\left[ {P(x) \Leftrightarrow Q(x)} \right] \Rightarrow \left( {\left( {\forall x} \right)\left[ {P(x)} \right] \Leftrightarrow \left( {\forall x} \right)\left[ {Q(x)} \right]} \right)$
$\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \Leftrightarrow \left( {\left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]} \right)$