Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to learn more about "Rules of Inference" and their application, but one thing always confuses me, and that is simplification "not s and p" to "not s".

I have looked at some examples: page 18 page 16

And I simply dont understand how is it possible to reduce expression.

Any help is welcome.


share|cite|improve this question
up vote 3 down vote accepted

$\lnot s \not\equiv (\lnot s \land p),\;$ but it is the case that $\;\lnot s\;$ follows from $\;\lnot s \land p$.

"$\lnot s \land p$ is true" means

  • $\lnot s$ is true, and $ p $ is true.

So it certainly follows that

  • "$\lnot s$ is true,"

just as it follows that

  • $p$ is true.

More simply put, we have

  • $\lnot s$ AND $p$.
    • Therefore $\lnot s$.
    • Therefore $p$.
share|cite|improve this answer
So when I have this type of assingments, where I need to rules of inference, I can presume that all of the expressions are true, and do assingment from there. Right? How do I know is it true or not? – depecheSoul Oct 26 '13 at 15:11
@amWhy: Right up your alley! =1 – Amzoti Oct 27 '13 at 0:07

This simplification is not intended to be in the sense of rewriting a nicer-looking, equivalent statement. Instead, the simplification is a nicer-looking statement that logically follows from the first. Of course, if $\neg s\vee p$ is true, then $\neg s$ is true (and also $p$ is true, so $p$ would be another simplification in this sense).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.