Problem 9 - Applied Abstract Algebra

Question

Find the conjunctive normal form of \(\left(x_{1}+x_{2}+x_{3}\right)\left(x_{1} x_{2}+x_{1}^{\prime} x_{3}\right)^{\prime} .\)

Step-by-Step Solution

Answer

The conjunctive normal form of \((x_1 + x_2 + x_3)(x_1 x_2 + x'_1x_3)'\) is \(x'_1x'_2 + x'_2x'_3 + x'_3\).

Step 1: Expand the brackets

Expand the brackets by multiplying each term inside the first brackets with the whole second brackets using distributive property. The new expression becomes:\[ (x_1 + x_2 + x_3)(x_1x_2)'(x’_1x_3)'.\]

Step 2: Apply De Morgan’s laws

De Morgan's laws state that the negation of a disjunction is the conjunction of the negations, and the negation of a conjunction is the disjunction of the negations. We can apply these properties to the expression as follows:\[ x'_1x_2'x'_1 x'_3 + x'_2x'_1 x'_3 + x'_3x'_1 x'_3.\]

Step 3: Simplify the Boolean expression

Here we can see the term \(x'_1x'_1\) and \(x'_3x'_3\) are repeating, according to the idempotent law in Boolean algebra, \(x.x = x\) and \(x+x=x\), we can simplify the expressions to:\[ x'_1x'_2 + x'_2x'_3 + x'_3.\]

Step 4: Conjunctive Normal Form

Finally, we've arrived at the CNF of the given Boolean expression. The CNF is a conjunction of disjunctions. Each conjunct is a disjunction of literals (in this case \(x'_1, x'_2\) and \(x'_3\)).

Key Concepts

De Morgan’s Laws

De Morgan's Laws are fundamental principles in Boolean Algebra that relate to complementing expressions.
They state that the complement of a conjunction (AND operation) is equivalent to the disjunction (OR operation) of the complements, and vice versa. This means:

• The negation of a conjunction: \((A \cdot B)' = A' + B'\)
• The negation of a disjunction: \((A + B)' = A' \cdot B'\)

These laws are crucial when simplifying logical expressions, as they allow us to transform the structure of expressions while retaining the same logical meaning. For example, if you encounter \((x_1x_2)'\), according to De Morgan’s Laws, it becomes \(x'_1 + x'_2\). This helps break down complex expressions into simpler parts, especially useful when moving an expression to a Conjunctive Normal Form (CNF).

Boolean Algebra

Boolean Algebra is the mathematical framework used to manipulate logical expressions.
It's built on binary variables that can take values of 0 (false) or 1 (true), allowing us to model and simplify logical statements effectively.
Here are some key properties of Boolean Algebra:

• Idempotent Law: \(x \cdot x = x\) and \(x + x = x\). A variable ANDed or ORed with itself is the variable.
• Complement Law: \(x + x' = 1\) and \(x \cdot x' = 0\). A variable ORed with its complement equals 1; ANDed equals 0.
• Identity Law: \(x + 0 = x\) and \(x \cdot 1 = x\). Zero is the identity for OR; one is the identity for AND.
• Distributive Law: Explored in detail later.

These rules allow expressions to be rewritten in a simpler format, aiding in tasks like finding the Conjunctive Normal Form.

Distributive Property

The Distributive Property is a key tool in Boolean Algebra and arithmetic.
It allows you to redistribute terms in an equation to simplify or rearrange them into a more usable form.
In Boolean Algebra, the Distributive Property works similarly to arithmetic but has unique characteristics:

• \(A \cdot (B + C) = (A \cdot B) + (A \cdot C)\): Multiplication over addition.
• \(A + (B \cdot C) = (A + B) \cdot (A + C)\): Addition over multiplication.

This lets you break down complex expressions into simpler parts. In logical expressions, applying the distributive property helps in reformatting expressions into Conjunctive Normal Form, by making it possible to assemble disjunctions into conjunctions or vice versa, fitting the rules of CNF.