# Use of Quantifiers

• Mar 24th 2013, 01:26 PM
rhymin
Use of Quantifiers
Another example from my textbook that I can't quite figure out. Let p(x), q(x) denote the following open statements.

p(x): x ≤ 3
q(x): x + 1 is odd

If the universe consists of all integers, what are the truth values of the following statements?

a) q(1)
b) ¬p(3)
c) p(7) v q(7)
d) p(3) ^ q(4)
e) ¬(p(-4) v q(-3))
f) ¬p(-4) ^ ¬q(-3)
• Mar 24th 2013, 01:35 PM
Plato
Re: Use of Quantifiers
Quote:

Originally Posted by rhymin
Another example from my textbook that I can't quite figure out. Let p(x), q(x) denote the following open statements.
p(x): x ≤ 3
q(x): x + 1 is odd

If the universe consists of all integers, what are the truth values of the following statements?
a) q(1)
b) ¬p(3)
c) p(7) v q(7)
d) p(3) ^ q(4)
e) ¬(p(-4) v q(-3))
f) ¬p(-4) ^ ¬q(-3)

We do not do your homework for you.

For c) $\displaystyle 7\not\le 3$ and $\displaystyle 7+1\text{ is not odd}.$ So is c) true or false? WHY?
• Mar 24th 2013, 01:40 PM
rhymin
Re: Use of Quantifiers
This isn't homework. I am taking a math class in about a month and am studying a textbook that I believe will help. Am I posting in the right forum? The questions I am posting are ones that didn't have great examples and are confusing to me.
• Mar 24th 2013, 01:44 PM
rhymin
Re: Use of Quantifiers
c) 7 is not less than or equal to 3 and 8 is not odd. so since both are false, it is false.

What happens if one is true and one is false?
• Mar 24th 2013, 01:54 PM
Plato
Re: Use of Quantifiers
Quote:

Originally Posted by rhymin
c) 7 is not less than or equal to 3 and 8 is not odd. so since both are false, it is false.
What happens if one is true and one is false?

That is not the question! It has nothing to do with it.

c) Asks if $\displaystyle [p(7) \vee q(7)$ is a true statement? When is an or, disjunction, true?
• Mar 24th 2013, 02:10 PM
rhymin
Re: Use of Quantifiers
It is not a true statement. I'm just not sure what "truth values" means.
• Mar 24th 2013, 02:19 PM
Plato
Re: Use of Quantifiers
Quote:

Originally Posted by rhymin
I'm just not sure what "truth values" means.

There are only two truth values: TRUE$\displaystyle (T)$ or FALSE$\displaystyle (F)$.

In this question $\displaystyle p(0)$ has a truth value of $\displaystyle T$ and $\displaystyle p(4)$ has truth value of $\displaystyle F$.
• Mar 25th 2013, 01:11 AM
rhymin
Re: Use of Quantifiers
Okay I learned a bit more about these symbols.
Are these right?

b) ¬p(3): not 3 ≤ 3 = False

c) p(7) v q(7): 7 ≤ 3 or 7 + 1 is odd...both are false, so it is False

d) p(3) ^ q(4): 3 ≤ 3 and 4 + 1 is odd...one is true, one is false, so it is False (since both must be true)

e) ¬(p(-4) v q(-3)): ¬(-4 ≤ 3 or -3 + 1 is odd)...one is false, one is true, so it is True

f) ¬p(-4) ^ ¬q(-3): ¬ -4 ≤ 3 and ¬ -3 + 1 is odd...one is false, one is true, so it is False
• Mar 25th 2013, 03:26 AM
Plato
Re: Use of Quantifiers
Quote:

Originally Posted by rhymin
Okay I learned a bit more about these symbols.
Are these right?

b) ¬p(3): not 3 ≤ 3 = False

c) p(7) v q(7): 7 ≤ 3 or 7 + 1 is odd...both are false, so it is False

d) p(3) ^ q(4): 3 ≤ 3 and 4 + 1 is odd...one is true, one is false, so it is False (since both must be true)

e) ¬(p(-4) v q(-3)): ¬(-4 ≤ 3 or -3 + 1 is odd)...one is false, one is true, so it is True

f) ¬p(-4) ^ ¬q(-3): ¬ -4 ≤ 3 and ¬ -3 + 1 is odd...one is false, one is true, so it is False

d) is true because both are true.
• Mar 25th 2013, 10:15 AM
rhymin
Re: Use of Quantifiers
Wow, I'm not sure how I missed that lol. What about (e)? Is that one also True? Doesn't just 1 have to be true with the "v" (or) symbol for it to be true? Or am I off there?
• Mar 25th 2013, 11:46 AM
HallsofIvy
Re: Use of Quantifiers
Quote:

e) ¬(p(-4) v q(-3)): ¬(-4 ≤ 3 or -3 + 1 is odd)...one is false, one is true, so it is True
You are missing the "not" at the start aren't you?
"$\displaystyle -4\le 3$" is true, "-3+ 1 is odd" is false so the statement "one of p(-4) OR q(3) is true" is true which means that "NOT....." is false.
• Mar 25th 2013, 12:39 PM
rhymin
Re: Use of Quantifiers
Ohhh i see now. Thank you so much for that. What I was doing was saying the first is true and second is false, and then just switching them. But I see that you are supposed to conclude that the entire statement is either true or false, and then use the negation symbol.
• Mar 26th 2013, 08:53 AM
Nehushtan
Re: Use of quantifiers
NB: Statements (e) and (f) are logically equivalent: they are either both true or both false.