# Thread: Writing propositions using connectives and quantifiers

1. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by MichaelH
Would I need to define Q at all?
Yes. Q is exactly what I wrote: Q(x,y) = "there is a real number between x and y with the property P". If you already created notation for that, then you did define Q and you should replace Q with the notation you created. Q is simply a placeholder for another expression.

2. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by SlipEternal
Yes. Q is exactly what I wrote: Q(x,y) = "there is a real number between x and y with the property P". If you already created notation for that, then you did define Q and you should replace Q with the notation you created. Q is simply a placeholder for another expression.
Thanks for clearing that up!

Sorry to be a pain but I'm also attempting another question to ensure I get this concept around my head. It states that:

For every natural number n, if n has the property P, then so does every smaller natural number

So I have started off with just trying to get the logic down: so it is saying that for all n in N, P(n) implies that P(n) occurs for all numbers less than n. Although this does not makes sense to me since the set is "EVERY natural number" so how is there a smaller natural number? This is the sort of logic that I really struggle with.

3. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by MichaelH
Thanks for clearing that up!

Sorry to be a pain but I'm also attempting another question to ensure I get this concept around my head. It states that:

For every natural number n, if n has the property P, then so does every smaller natural number

So I have started off with just trying to get the logic down: so it is saying that for all n in N, P(n) implies that P(n) occurs for all numbers less than n. Although this does not makes sense to me since the set is "EVERY natural number" so how is there a smaller natural number? This is the sort of logic that I really struggle with.
It is saying that if n has the property P, then 1 has the property, 2 has the property, ..., n-2 has the property, and n-1 has the property. Those are all of the natural numbers that are smaller than n. So, for every natural number, check to see if it has the property. If it does, then every natural number less than that also has the same property.

Now, are you looking for a mathematical expression that states what you are given? Or could it be any logically equivalent expression?

For example: "For all natural numbers m and n, if m is less than n and n has the property p, then m has the property p." That is logically equivalent to the statement, "For every natural number n, if n has the property P, then so does every smaller natural number"

4. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by SlipEternal
It is saying that if n has the property P, then 1 has the property, 2 has the property, ..., n-2 has the property, and n-1 has the property. Those are all of the natural numbers that are smaller than n. So, for every natural number, check to see if it has the property. If it does, then every natural number less than that also has the same property.

Now, are you looking for a mathematical expression that states what you are given? Or could it be any logically equivalent expression?

For example: "For all natural numbers m and n, if m is less than n and n has the property p, then m has the property p." That is logically equivalent to the statement, "For every natural number n, if n has the property P, then so does every smaller natural number"
Ok, I am looking to write the proposition using connectives and quantifiers.

Could I write it as:

(∀n ∈ N)(∀m ∈ N)((m<n) ∧ P(n)) ---> P(m))

5. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by MichaelH
Ok, I am looking to write the proposition using connectives and quantifiers.

Could I write it as:

(∀n ∈ N)(∀m ∈ N)((m<n) ∧ P(n)) ---> P(m))
I gave you a logically equivalent expression. You gave me a correct mathematical statement for the expression I gave, but it is not the same expression you were originally given. Will your professor accept a logically equivalent expression? If so, then that is perfect. Otherwise, see if you can write something closer to the original proposition.

6. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by SlipEternal
I gave you a logically equivalent expression. You gave me a correct mathematical statement for the expression I gave, but it is not the same expression you were originally given. Will your professor accept a logically equivalent expression?
No I don't think so. I have to write out mathematical statement for the original statement.

7. ## Re: Writing propositions using connectives and quantifiers

So, try the original proposition. You will wind up with nested conditionals.

8. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by SlipEternal
So, try the original proposition. You will wind up with nested conditionals.
Hmmm I am not sure how to express it without introducing m with the equivalent statement, that is the problem that I have.

9. ## Re: Writing propositions using connectives and quantifiers

$\displaystyle (\forall n \in \mathbb{N})(P(n) \Rightarrow [(\forall m\in \mathbb{N})(m<n \Rightarrow P(m))])$

10. ## Re: Writing propositions using connectives and quantifiers

Originally Posted by MichaelH
For every natural number n, if n has the property P, then so does every smaller natural number.
You can save yourself some trouble if you first setup some preliminaries.
First specify the universe of discourse as $\displaystyle \mathbb{N}$.
Next define predicate functions:
$\displaystyle P(n)$ means "$\displaystyle n$ has property $\displaystyle P$".
$\displaystyle L(m,n)$ means that $\displaystyle "m<n$, $\displaystyle m\text{ is less than }n"$.

Now the translation of "For every natural number n, if n has the property P, then so does every smaller natural number."
is $\displaystyle \left( {\forall n} \right)\left( {\forall m} \right)\left[ {P(n) \wedge L(m,n) \to P(m)} \right]$.

You see there is no need to state that these natural numbers because that is our universe.

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