# Thread: What is the difference between these two logical statements?

1. ## What is the difference between these two logical statements?

Hi,

I'm teaching myself propositional logic at home and I can't understand the difference between these two statements in the book. I think one of these represents that 'age is not between 65 and 80 inclusive', but I'm really not sure. If someone could explain what each of the statements mean that would be great. Then I can carry on with my reading!!

Here are the two I don't know the difference between:

¬ ((age ≥ 65) ∨ (age ≤ 80))

¬ (age ≥ 65) ∨ ¬ (age ≤ 80)

2. ## Re: What is the difference between these two logical statements?

a)

$\displaystyle \lnot ((age \geq 65) \lor (age \leq 80))\Leftrightarrow \lnot(age \geq 65) \land \lnot(age \leq 80)\Leftrightarrow$

$\displaystyle \Leftrightarrow (age < 65) \land (age > 80)$ , so you have false statement.

b)

similar as case a)

3. ## Re: What is the difference between these two logical statements?

Originally Posted by thefurrycritter
Here are the two I don't know the difference between:

¬ ((age ≥ 65) ∨ (age ≤ 80))

¬ (age ≥ 65) ∨ ¬ (age ≤ 80)
It is easy:
$\displaystyle \neg \left( {p \vee q} \right) \equiv \neg p \wedge \neg q\;\& \,\neg p \vee \neg q \equiv \neg (p \wedge q)$

4. ## Re: What is the difference between these two logical statements?

Originally Posted by thefurrycritter
¬ ((age ≥ 65) ∨ (age ≤ 80))

¬ (age ≥ 65) ∨ ¬ (age ≤ 80)
The first one means "it is not true that: either age is at least 65, or age is also at most 80". Or, put another way (see DeMorgan's laws on Wikipedia): "age is not at least 65, and age is not at most 80". We can simplify this noting that the inverse of "at least" is "less than", etc.: "age is less than 65, but more than 80". This is a contradiction.

The second one is equivalent to "age is less than 65 or more than 80", which is the opposite of what you want (this is true only when age is not in the range you want).

Just build it by using English. "Age is between 65 and 80 inclusive," becomes "age is at least 65 and age is at most 80". Does that help...?

5. ## Re: What is the difference between these two logical statements?

Originally Posted by Annatala
The first one means "it is not true that: either age is at least 65, or age is also at most 80". Or, put another way (see DeMorgan's laws on Wikipedia): "age is not at least 65, and age is not at most 80". We can simplify this noting that the inverse of "at least" is "less than", etc.: "age is less than 65, but more than 80". This is a contradiction.

The second one is equivalent to "age is less than 65 or more than 80", which is the opposite of what you want (this is true only when age is not in the range you want).

Just build it by using English. "Age is between 65 and 80 inclusive," becomes "age is at least 65 and age is at most 80". Does that help...?
I'm very new to this so I'm not familiar with that many symbols, but I think it makes a bit more sense now. So just to clarify, do none of the two statements mean that the 'age is NOT between 65 and 80'?

Thanks again

6. ## Re: What is the difference between these two logical statements?

The first statement is false. The second statement means that age is less than 65 or greater than 80 , so it can't be between 65 and 80

7. ## Re: What is the difference between these two logical statements?

Originally Posted by princeps
The first statement is false. The second statement means that age is less than 65 or greater than 80 , so it can't be between 65 and 80
ah ok, thanks. So, if you were to make a statement using the less than, more than etc and other prop logic symbols to represent the condition that age is not between 65 and 80 inclusive , what would that look like?

Many thanks again (I'm just trying to learn!)

8. ## Re: What is the difference between these two logical statements?

Originally Posted by thefurrycritter
ah ok, thanks. So, if you were to make a statement using the less than, more than etc and other prop logic symbols to represent the condition that age is not between 65 and 80 inclusive , what would that look like?
Your second example, ¬ (age ≥ 65) ∨ ¬ (age ≤ 80), is just such a statement. You could write it more simply as:

(age < 65) ∨ (age > 80)

...or even as:

¬ (age ≥ 65 $\displaystyle \wedge$ age ≤ 80).

All of those mean the same thing: age is either less than 65 or greater than 80.