Writing propositions using connectives and quantifiers

Quote:

---

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.

Quote:

---

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.

Quote:

---

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"

Quote:

---

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))

Quote:

---

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.

Quote:

---

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.

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

Quote:

---

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.

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

Quote:

---

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.