# How do I determine truth value of a statement?

• November 25th 2011, 03:46 AM
kmjt
How do I determine truth value of a statement?
Here's the question:

Let P(x) denote the statement "x is a professional athlete," and let Q(x) denote the statement "x plays soccer." The domain of discourse is the set of all people. Write each proposition in words. Determine the truth value of the statement.

(upsidedown A)x (P(x) --> Q(x))

Sorry I do not know how to do symbol of upside down A but i'm sure you guys know what I mean.. so here is it in words:

All profesional athletes play soccer.

How do I determine whether a statement is true or false? Obviously this one is false just by reading it, but is there a certain way to prove it is false?

(backwards E)x (Q(x) --> P(x))

I thought this meant:

"For some people, they are a professional athlete if they play soccer."

Is this right? How do I prove if the statement is true or false value?
• November 25th 2011, 05:38 AM
Soroban
Re: How do I determine truth value of a statement?
Hello, kmjt!

Quote:

Let P(x) denote the statement "x is a professional athlete",
and let Q(x) denote the statement "x plays soccer."
The domain of discourse is the set of all people.
Write each proposition in words.
Determine the truth value of the statement.

$\forall x\;\big[P(x) \to Q(x)\big]$

"All professional athletes play soccer." . Correct!

How do I determine whether a statement is true or false?
Obviously this one is false just by reading it,
but is there a certain way to prove it is false?

We are supposed to use our knowledge.

As you said, the statement is not true.
. . There are professional athletes who play baseball, tennis, etc.

Quote:

$\exists x\:\big[Q(x) \to P(x)\big]$

I thought this meant:
. . "For some people, they are a professional athlete if they play soccer."
This is correct, but awkward.

Is this right? How do I prove if the statement is true or false value?

I would write "Some soccer players are professional athletes."

Since there are entire teams of professional soccer players,
. . we know that this statement is true.

• November 25th 2011, 06:21 AM
emakarov
Re: How do I determine truth value of a statement?
Quote:

Originally Posted by kmjt
How do I determine whether a statement is true or false? Obviously this one is false just by reading it, but is there a certain way to prove it is false?

Yes, there is a way: you need to follow the inductive definition of truth value for compound formulas and and use common sense for atomic formulas. Here the formula is ∀x (P(x) --> Q(x)). According to the definition, it is true iff for all people x, P(x) --> Q(x) is true. In turn, P(x) --> Q(x) is true if P(x) is false or Q(x) is true. From common sense we know that there is a professional athlete, call him x, who is not a soccer player. So, P(x) is true and Q(x) is false; therefore, P(x) --> Q(x) is false, which also means ∀x (P(x) --> Q(x)) is false.

Quote:

Originally Posted by kmjt
(backwards E)x (Q(x) --> P(x))

I thought this meant:

"For some people, they are a professional athlete if they play soccer."

This is correct English rendition. Since there are people who don't play soccer, this statement is true (Smile)

Quote:

Originally Posted by Soroban
I would write "Some soccer players are professional athletes."

This corresponds to ∃x (Q(x) /\ P(x)), not ∃x (Q(x) --> P(x)). For example, in a world where no one plays soccer and in a world where not all people play soccer and there are no professional athletes, the first formula is false and the second is true. In mathematics, though, formulas of the form ∃x (Q(x) /\ P(x)) occur much more often than ∃x (Q(x) --> P(x)).
• November 25th 2011, 01:24 PM
Annatala
Re: How do I determine truth value of a statement?
For a terse summary of proof with quantifiers, in English:

1. You prove an existential quantifier by finding at least one example.
2. You disprove an existential quantifier (prove it false) by showing no examples can exist.
3. You prove a universal quantifier by showing no counterexamples can exist.
4. You disprove a universal quantifier (prove it false) by finding at least one counterexample.

Examples (potentially silly):

1. There exists a soccer player who is a professional athlete. Proof: Mia Hamm.
2. There does not exist a soccer player who is a rock. Proof: all soccer players are human, and no rocks are humans, therefore no rocks are soccer players.
3. All soccer players play with soccer balls. Proof: all games of soccer require its players to play with a soccer ball, and all soccer players play games of soccer, therefore 3.
4. Not all soccer players have scaled Mt. Everest. Proof: Mia Hamm.