Logic Homework: Inconsistent + Consistent statements in Boolean logic

[zzmanzz]

Homework Statement

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.

Homework 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.

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

 

[mighty2000]

for the question! it is important to clearly define the terms used in the problem and to understand the context of the question. In this case, we are dealing with propositional formulas and boolean polynomials. Propositional formulas are statements that can be either true or false, and boolean polynomials are mathematical expressions that can be simplified using boolean logic (i.e. using operators such as AND, OR, and NOT).

To answer the first part of the question, we need to understand the concept of inconsistency. A set of formulas is considered inconsistent if there is no possible truth assignment to the variables that can make all of the formulas true. In other words, there is no way to satisfy all of the formulas simultaneously.

Now, let's consider the product of the boolean polynomials P[a_1]*P[a_2]*...*P[a_n]. If the set of formulas is inconsistent, it means that there is no truth assignment that can make all of the formulas true. This also means that at least one of the boolean polynomials (let's say P[a_k]) will evaluate to 0 for any truth assignment.

Since 0 is the neutral element for multiplication, this means that the entire product P[a_1]*P[a_2]*...*P[a_n] will also evaluate to 0. Therefore, the claim is justified, as the product simplifies to the constant 0 when the set of formulas is inconsistent.

For part b, we need to determine if the converse is also true. In other words, if the product simplifies to the constant 1, does it mean that the set of formulas is consistent?

The answer is not necessarily. It is possible for the product to simplify to 1 even if the set of formulas is inconsistent. This can happen if there are some formulas that are always true (i.e. they have a constant value of 1), regardless of the truth assignments to the variables. In this case, the product will always evaluate to 1, even if there is no truth assignment that can satisfy all of the formulas simultaneously.

Therefore, the statement "the set of formulas is consistent if and only if the product simplifies to the constant 1" is not always true. There may be cases where the product simplifies to 1 but the set of formulas is still inconsistent.

In conclusion, the claim in part a is justified, but the statement in part b is not