Problem 54 - Discrete Mathematics with Applications

Question

Simplify each boolean expression. $$(p \wedge \sim q) \vee(\sim p \wedge q) \vee(\sim p \wedge \sim q)$$

Step-by-Step Solution

Answer

The simplified boolean expression is \(1\).

Step 1: Commutative Law

Using the Commutative Law, we will rearrange the terms in the expression to group together similar terms: \((p \wedge \sim q) \vee(\sim p \wedge q) \vee(\sim p \wedge \sim q) = (\sim q \wedge p) \vee (q \wedge \sim p) \vee (\sim q \wedge \sim p)\)

Step 2: Distributive Law

Next, we'll apply the Distributive Law to combine the first two terms: \((\sim q \wedge p) \vee (q \wedge \sim p) = (p \wedge (\sim q \vee q)) \vee (\sim p \wedge (\sim q \vee q))\)

Step 3: Identity Law

Using the Identity Law, where \(x \vee \sim x = 1\), we can simplify the expression: \((p \wedge (\sim q \vee q)) \vee (\sim p \wedge (\sim q \vee q)) = (p \wedge 1) \vee (\sim p \wedge 1)\)

Step 4: Annulment Law

By applying the Annulment Law, where \(x \wedge 1 = x\), we can further simplify the expression: \((p \wedge 1) \vee (\sim p \wedge 1) = p \vee \sim p\)

Step 5: Identity Law

Finally, using the Identity Law again, we can simplify the expression to a constant value: \(p \vee \sim p = 1\) The simplified boolean expression is: $$1$$

Key Concepts

Commutative Law

The Commutative Law in Boolean algebra is quite similar to its arithmetic counterpart. It states that the order of the operands does not change the result of the operation. For boolean expressions, it applies to both AND () and OR () operations.

• For the OR operation: \( A \vee B = B \vee A \)
• For the AND operation: \( A \wedge B = B \wedge A \)

This law allows us to rearrange terms in a boolean expression, which can be crucial for simplification. In the original problem, the expression was rearranged using the Commutative Law to group similar terms together. This property helps us see potential simplifications more clearly by placing terms in a more manageable order.

Distributive Law

The Distributive Law is a very helpful tool in Boolean algebra, allowing us to distribute one operation over another. It is slightly different from its arithmetic counterpart. In Boolean algebra, the distributive property allows you to handle expressions involving both AND () and OR () operations.

• \( A \wedge (B \vee C) = (A \wedge B) \vee (A \wedge C) \)
• \( A \vee (B \wedge C) = (A \vee B) \wedge (A \vee C) \)

In the exercise, the Distributive Law was used to simplify the expression by combining like terms. For instance, in the step, the terms were organized into a simpler format using this property. Your goal when applying the Distributive Law is to break down complex expressions into more simple or familiar parts, paving the way for further simplifications.

Identity Law

The Identity Law in Boolean algebra involves the principle that an operation with a particular identity element leaves the initial value unchanged. In these operations:

• For OR: \( A \vee 0 = A \)
• For AND: \( A \wedge 1 = A \)

In Boolean simplification, recognizing and applying identity laws can lead to much simpler and cleaner expressions. In the given solution, the Identity Law was used multiple times. One key application was recognizing expressions like \( p \vee \sim p = 1 \), simplifying it to a constant value. Such simplifications specifically take advantage of the fact that any variable ORed with its complement results in 1, which is a crucial identity rule in Boolean algebra.

Annulment Law

The Annulment Law is a fundamental concept in Boolean algebra. It describes how certain operations result in a definitive outcome, regardless of the other operand.

• For OR: \( A \vee 1 = 1 \)
• For AND: \( A \wedge 0 = 0 \)

In the solution, the Annulment Law was applied to break down the expression significantly. This law specifically comes into play when an expression is simplified to a single term or constant. As seen in the exercise, terms like \( p \wedge 1 \) simplify directly to \( p \) itself, showcasing the practical use of this fundamental law. By systematically applying the Annulment Law along with other laws, we achieve a minimal expression that is easy to interpret and apply in logical circuits.