How to prove that this argument is valid?

Question

I've been stuck on this problem for a while, mainly because I'm not sure if I'm using predicate symbols correctly , and partyl because I can't figure out how to get to the conclusion. Ive done questions like this before , but for some reason i just cant figure this out. I hate to ask , but please help!

Every truck has a radio
Some trucks have a CD player
Therefore some trucks have both a radio and a cd player.

I believe the following to be the starting point.

forall xT(x) implies R(x)
thereexists xT(x) implies C(x)

Answer 1

Your choice of symbols is fine:

$T(x)$: $x$ is a truck

$R(x)$: $x$ has a radio

$C(x)$: $x$ has a CD player

Your symbolization of the first statement is also correct:

$\forall x (T(x) \rightarrow R(x))$ : All trucks have a radio

But your symbolization of the second statement was not correct. You said:

$\exists x (T(x) \rightarrow C(x))$ : Some trucks have a CD player

Your symbolization reads: there is something such that if it is a truck, then it has a CD player. But notice that if there is anything in the world that is not a truck, then that very non-truck object will make that statement true, since the antecedent of the conditional will be false, and that will make the whole conditional true (check its truth-table: if the 'if' part is false, any 'if ... then' statement is automatically true). So: your symbolized sentence could be true without any trucks having any CD player, so this is not what you want.

Long story short: you really don't want to use a conditional in combination with an existential: that is almost always asking for trouble. Instead, use an $\land$. So, the second statement should be:

$\exists x (T(x) \land C(x))$

To symbolize the conclusion you do a similar thing:

$\exists x (T(x) \land (C(x) \land R(x))$ Some trucks have both a radio and a CD player

OK, so as a whole argument, you have:

$\forall x (T(x) \rightarrow R(x))$

$\exists x (T(x) \land C(x))$

$\therefore \exists x (T(x) \land (C(x) \land R(x))$

Now, how do you prove this valid? It really depends on what system you are using (and hence what inference rules are available for you to use), but a typical proof would go something like this:

1. $\forall x (T(x) \rightarrow R(x))$ (premise)

2. $\exists x (T(x) \land C(x))$ (premise)

3. $T(a) \land C(a)$ Existential Elimination 2

4. $T(a)$ Simplification 3

5. $T(a) \rightarrow R(a)$ Universal Elimination 1

6. $R(a)$ Modus Ponens 4,5

7. $C(a)$ Simplification 3

8. $C(a) \land R(a)$ Conjunction 6,7

9. $T(a) \land (C(a) \land R(a))$ Conjunction 5,8

10. $\exists x (T(x) \land (C(x) \land R(x))$ Existential Introduction 9

Answer 2

$$\forall\ t \in T: R(t)$$ and $$\exists\ t\in T: C(t)$$ implies $$\exists\ t\in T: R(t)\land C(t).$$

Proof: choose some $t$ such that $C(t)$; by the first predicate, $R(t)$ holds. Then $R(t)\land C(t)$ as well.