Understanding the Distribution of Negation in Propositional Logic

[sunny79]

Homework Statement

Let P stand for "I'll buy the pants.
Let S stand for "I'll buy the shirt.
What english sentence is represented by the formula
~(P ^ ~S)

Relevant Equations

~(P ^ ~S)

Given that the negation is distributed across parenthesis, P become ~p and S gets double negation ~~S. Hence my solution was " I will not buy the pants but I will buy the shirt. (or and I will buy the shirt, since but can be used in the place of and).

This is from How to prove things by Velleman 3rd edition, chapter 1, section 1.1. However, at the back the solution said I will not buy the pants without buying the shirt. That left me confused.

Look forward to the input.

 

[Stephen Tashi]

(or and I will buy the shirt, since but can be used in the place of and).

You haven't stated the rule correctly. The "and" must be changed to "or".

$\lnot( A \land B ) \iff \lnot A \lor \lnot B$

 

[sunny79]

Hello Stephen! They provided the equation outlined below. My task was to convert it into english.
¬(P ^ ¬S). My shot at the solution was " I won't buy pants but I will buy shirt." The answer in the back stated " I won't buy pants without the shirt".

 

[Mark44]

This problem might be an application of one of DeMorgan's Laws; namely, that ¬(P ^ ¬S) ⇔ ¬P ∨ S. The literal translation to English would be "I won't buy the pants or I will buy the shirt."
This proposition would be true in any of the following situations:
1. I don't buy the pants, and I buy the shirt. (both clauses true)
2. I don't buy the pants, and I don't buy the shirt. (first clause true, second clause false)
3. I buy the pants, and I buy the shirt. (first clause false, second clause true)
It's certainly the case that "I won't buy pants without the shirt," as the book's answer shows, but it seems to me that there are scenarios that this answer omits.

The proposition would be false if both clauses are false.
I buy the pants and I don't buy the shirt.

 

[Stephen Tashi]

¬(P ^ ¬S). My shot at the solution was " I won't buy pants but I will buy shirt."

Your solution is equivalent to $\lnot P \land S$. This is incorrect. As I mentioned before, the correct (intermediate) answer is $\lnot P \lor S$ (using $\lor$ instead of $\land$).

An English equivalent of "I won't buy the pants without the shirt" is "If I buy the pants, I will buy the shirt". To proceed from the intermediate answer to an if...then phrasing use:

$(\lnot A \lor B) \iff (A \implies B)$

 

[sunny79]

@Stephen Tashi @Mark44 That helps a lot. Thank you. :)