# DISCRETE MATH: Which of these compound propositions is satisfiable? No use truthtable

1. Jan 17, 2007

### VinnyCee

1. The problem statement, all variables and given/known data

Is this compound statement satisfiable?

$$(p\,\vee\,q\,\vee\,\neg\,r)\,\wedge\,(p\,\vee\,\neg\,q\,\vee\,\neg\,s)\,\wedge\,(p\,\vee\,\neg\,r\,\vee\,\neg\,s)\,\wedge\,(\neg\,p\,\vee\,\neg\,q\,\vee\,\neg\,s)\,\wedge\,(p\,\vee\,q\,\vee\,\neg\,s)$$

2. Relevant equations

I guess you are supposed to use the following instead of truth tables somehow:

Logical equivalences - Domination, Idempotent, Double negation, Commutative, De Morgan's, Absorption, Negation, Associate, Distributive.

3. The attempt at a solution

I "converted" the first term in the expression:

$$(p\,\vee\,q\,\vee\,\neg\,r)\,\equiv\,\left[(\neg\,p\,\longrightarrow\,q)\,\vee\,\neg\,r\right]$$

Now what do I do though?

2. Jan 18, 2007

### VinnyCee

I figured it out!

It was simple.

The equation is satifiable.

Set p = TRUE and q = FALSE. Since it is all OR logical connectives, it was simple to find these values which make the statement true.

3. Jan 18, 2007

### matt grime

I really hope you didn't just try things at random. Just use those laws above. It is quite straight forward.

4. Mar 12, 2010