Re: Sentential Logic Problem

Quote:

Originally Posted by

**Turgus** Hi guys,

Having issues understanding the answer provided by the book to the following question:

Question: Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentence is represented by the following expression

$\displaystyle \neg(P \land \neg S)$

The books answer is: I won't buy the pants without the shirt.

My answer turns out to be: I will not buy the pants or I will buy the shirt.

To get my answer I did the following:

$\displaystyle \neg(P \land \neg S)$

$\displaystyle \neg P \lor \neg \neg S$ (De Morgan's Law)

$\displaystyle \neg P \lor S$ (Double Negation Law)

But your answer has exactly the same meaning as the one in the textbook.

$\displaystyle \neg(P \land \neg S)$ is literally read "**It false that I buy the pants **__and not__ buy the shirt.

Re: Sentential Logic Problem

Quote:

Originally Posted by

**Plato** But your answer has exactly the same meaning as the one in the textbook.

$\displaystyle \neg(P \land \neg S)$ is literally read "**It false that I buy the pants **__and not__ buy the shirt.

I don't see the relation how my answer has the exact same meaning as the one in the textbook....

The books answer to me is saying:

If I buy the shirt, I will buy the pants.

OR

If I don't buy the shirt, I won't buy the pants.

Isn't that a completely different meaning from my answer?

Re: Sentential Logic Problem

Quote:

Originally Posted by

**Turgus** I don't see the relation how my answer has the exact same meaning as the one in the textbook....

The books answer to me is saying:

If I buy the shirt, I will buy the pants.

OR

If I don't buy the shirt, I won't buy the pants.

Isn't that a completely different meaning from my answer?

Well, you just don't understand the "if,,,then..."

It is the case that $\displaystyle P \to S \equiv \,\neg P \vee S \equiv \,\neg \left( {P \wedge \neg S} \right)$.

"If I buy the pants then I buy the shirt" is the same as "I don't buy the pants or I buy the shirt" is the same as "it is false that I buy the pants and not buy the shirt".

For me those have the meaning as "I won't buy the pants without the shirt".

Re: Sentential Logic Problem

Quote:

Originally Posted by

**Turgus** I don't see the relation how my answer has the exact same meaning as the one in the textbook....

The books answer to me is saying:

If I buy the shirt, I will buy the pants.

OR

If I don't buy the shirt, I won't buy the pants.

Isn't that a completely different meaning from my answer?

Here is the deal:

The book answer is correct as also yours.

The meaning of the $\displaystyle \neg(P \land \neg S)$ is that:

I will not (buy the pants and will not buy the shirt), i.e:

I will not buy the pants and then i will not buy the shirt also, or:

If i will buy the pants then i will buy the shirt also.

The above correct answer is the same (obviously) with that of the book.

Your answer says:

The answer is : $\displaystyle \neg P \lor S$ And that means either and exactly one of the above 3 is true:

a)I will **not buy the pants **and will not buy the shirt

b)I will buy the shirt and will buy the pants

c)I will **not buy the pants** AND will buy the shirt.

Since there are 4 combinations of buying/not buying shirts/pants, the extra option for us to do is:

d)I will buy the pants and not buy the shirt.

Now:

The book answer says that "If i **will buy the pants**, i will buy the shirt also".

So your a) and c) do not contradict with what the book's answer says. **(1)**

And of course your b) does also not contradict the book's answer. **(2)**

Note also that a), b), c), d) are all contradicting to each other. **(3)**

Note also a)+b)+c)+d) make all our possible cases. **(4)**

So **(1)+****(2)+****(3)+****(4)** mean that the book's answer is completely equivalent to yours.

Re: Sentential Logic Problem

Thanks guy! I understand it now.

Re: Sentential Logic Problem

Quote:

Originally Posted by

**Turgus** Thanks guy! I understand it now.

Yes, it's important to understand that an implication of the form A => B means that:

A is true and B is true

A is false and B is true

A is false and B is false.

So the original implication of:

"If I buy the pants then I buy the shirt" , means:

I will buy the pants and i will buy the shirt.

I will not buy the pants and i will buy the shirt.

I will not buy the pants and i will not buy the shirt.