# Logic Homework: Inconsistent + Consistent statements in Boolean logic

1. Oct 21, 2013

### zzmanzz

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

Let a_1, a_2,...a_n be propositional formulas.

Let P[a_1],P[a_2], ..,P[a_n] be the boolean polynomials associated with a_i for i = 1..n

Compute the simplest form of the product P[a_1]*P[a_2]*...*P[a_n] as a Boolean polynomial.

Claim: The set of formulas a_1,a_2,...,a_n is inconsistent if and only if the product simplifies to the constant 0.

a) Justify this claim.
b) Is is it true: that if a_1,a_2,...,a_n is consistent if and only if the product simplifies to the constant 1.

2. Relevant equations

Inconsistent statement:Let a_1,...,a_n be a collection of propositional formulas. It is inconsistent if no truth assignments to the variables in the propositional formulas which turns all of the formulas simultaneously true.

Consistent: There exists a truth assignment to the variables in the propositional formulas which turns them all true.

Simplest Boolean polynomial form: The polynomial can't be simplified any further. I.e. P = xyz is the simplest form. P = x^2 can be reduced to P = x.

3. The attempt at a solution

The claim makes sense because even if one propositional formula is false,the entire product would equal to 0. But this is with plugging the values into the propositional formula. Not what the question asks. Any idea on how I would start proving the claim a and proving/disproving b?

Thanks