# Show that two quantified statements are logically equivalent

1. Nov 26, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Show that $\forall x(P(x) \land Q(x)) \equiv \forall xP(x) \land \forall xQ(x)$

2. Relevant equations

3. The attempt at a solution
Based on my work from propositional logic, to show that two expressions are logically equivalent, then we have to show that $\forall x(P(x) \land Q(x)) \Longleftrightarrow \forall xP(x) \land \forall xQ(x)$ is a tautology; that is, it is always true. It is always true if they have the same truth values for all x in the domain. For propositional logic, it was a matter of listing out the finite combinations of truth values and showing that they are always the same. However, with predicate logic, we are dealing with an infinite domain of discourse, so we can't just list them off. How should I proceed then?

Last edited: Nov 26, 2016
2. Nov 26, 2016

### Stephen Tashi

What variable does $\forall$ quantify? Did you mean $( \forall x, P(x) \land Q(x)\ ) \equiv (\forall x, P(x))\land(\forall x, Q(x))$.

What assumptions and theorems have you studied in proposition logic ? "Universal generalization", "Existential instantiation" etc. ?

3. Nov 26, 2016

### Mr Davis 97

I fixed the errors in the original post.

I was thinking that maybe I could use universal generalization to show that, if the LHS is true, then $P(a) \land Q(a)$ is true for all a in the domain. Then by universal generalization, we would have the RHS. However, I don't see how this establishes that the biconditional is a tautology, which is necessary to show that they are logically equivalent.

4. Nov 26, 2016

### Stephen Tashi

A biconditional is just two conditionals. Show each separately.

5. Nov 26, 2016

### Mr Davis 97

If I show that LHS implies the RHS, and that the RHS implies the LHS, then I would show that one being true implies that the other is true. But in order for it to be a tautology, don't you have to also show that if one is false than the other must be false as well?

6. Nov 26, 2016

### Stephen Tashi

That will depend on how your text materials define the relation "$\equiv$".

Is there a theorem or definition in your materials that says: $( ( A \implies B) \land (B \implies A) ) \implies A \equiv B$ ?

7. Nov 27, 2016

### Staff: Mentor

It is always a good idea to start this way: What does it mean, if one side were wrong? Can the other still be true?
And then the other way around.