Question

For Boolean identities match the pair: (1) \(\underline{\underline{A}}\) (P) \(\underline{\mathrm{A}}+\underline{\mathrm{B}}\) (2) \(\underline{\mathrm{A}+\mathrm{B}}\) (Q) A \cdot B (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) (R) \(\underline{A} \cdot \underline{B}\) (4) \(\mathrm{A} \cdot(\underline{\mathrm{A}}+\mathrm{B})\) (S) \(\mathrm{A}\) (A) \((1)-(\mathrm{S}),(2)-(\mathrm{P}),(3)-(\mathrm{Q}),(4)-(\mathrm{R})\) (B) \((1)-(\mathrm{S}),(2)-(\mathrm{R}),(3)-(\mathrm{Q}),(4)-(\mathrm{P})\) (C) (1) - (S), (2) - (Q), (3) - (P), (4) - (R) (D) \((1)-(\mathrm{S}),(2)-(\mathrm{R}),(3)-(\mathrm{P}),(4)-(\mathrm{Q})\)

Short answer

The short answer is: (D) (1) - (S), (2) - (R), (3) - (P), (4) - (Q)

Step-by-step solution

1) Simplifying the given Boolean identities expressions

Let's start by simplifying or rearranging the expressions: (1) \(\underline{\underline{A}}\) : The complement of complement A is simply A. (2) \(\underline{\mathrm{A}+\mathrm{B}}\) : This is the complement of the sum of A and B. (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) : This is the complement of the product of A and B. (4) \(\mathrm{A} \cdot(\underline{\mathrm{A}}+\mathrm{B})\) : Simplifying this expression using the distributive law of Boolean algebra, we have \(A \cdot \underline{A} + A \cdot B\). And since \(A\cdot\underline{A}=0\), it becomes \(0 + A \cdot B\), which is simply \(A \cdot B\). Now let's move to the equivalent expressions.

2) Identify the equivalent expressions

For each original expression (1, 2, 3, and 4), we want to associate it with the equivalent expression (P, Q, R, and S): (P) \(\underline{\mathrm{A}}+\underline{\mathrm{B}}\) : Using De Morgan's law, this simplifies to \(\underline{A \cdot B}\). (Q) A \(\cdot\) B : This is simply the product of A and B. (R) \(\underline{A} \cdot \underline{B}\) : Using De Morgan's law, this simplifies to \(\underline{A + B}\). (S) A : This is simply A.

3) Match the pairs

Now that we have simplified and/or identified the equivalent expressions for each Boolean identity, we can match the pairs as follows: (1) A -> (S) A (2) \(\underline{\mathrm{A}+\mathrm{B}}\) -> (R) \(\underline{A \cdot B}\) (3) \(\underline{\mathrm{A} \cdot \mathrm{B}}\) -> (P) \(\underline{A} + \underline{B}\) (4) \(A \cdot B\) -> (Q) A \(\cdot\) B So the correct choice is given by option (D): (D) (1) - (S), (2) - (R), (3) - (P), (4) - (Q)

Key concepts

Complement of a Boolean Expression

Complementing a Boolean expression is a fundamental action in Boolean algebra, focused on reversing the trueness of a given proposition. If you have a Boolean expression, its complement is denoted by adding a bar or overline over the whole expression. Here are some key points to understand:

• The complement of true (\(1\)) is false (\(0\)), and the complement of false is true.
• For a single variable \(A\), its complement is written as \(\overline{A}\), which is false when \(A\) is true, and true when \(A\) is false.
• For more complex expressions, the complement negates each element and operation, basically flipping the ANDs to ORs and vice versa.

Understanding the complement concept helps simplify expressions and is crucial when applying Boolean laws such as De Morgan's to switch between conjunctions and disjunctions.

De Morgan's Laws

De Morgan's Laws are essential rules in Boolean algebra that detail how to simplify the complement of complex expressions. They are instrumental when you need to deal with negations of combined expressions. The laws are stated as follows:

• The complement of a conjunction (AND) is the disjunction (OR) of the complements: \(\overline{A \cdot B} = \overline{A} + \overline{B}\).
• The complement of a disjunction (OR) is the conjunction (AND) of the complements: \(\overline{A + B} = \overline{A} \cdot \overline{B}\).

These operations seem like opposites because they reverse the logic gate interactions. When applied, De Morgan's laws allow the transformation of an entire expression by breaking it down into smaller, simpler parts that are easier to manage or compute, especially in digital circuits and logic proofs.

Distributive Law in Boolean Algebra

The distributive law is a critical principle in Boolean algebra that outlines how expressions can be expanded or factored. It is similar to the distribution of multiplication over addition in traditional algebra but adapted for logical operations. Let's understand the key aspects of this law:

• The law can distribute an AND operation across an OR: \(A \cdot (B + C) = (A \cdot B) + (A \cdot C)\).
• Conversely, an OR can be distributed over an AND: \(A + (B \cdot C) = (A + B) \cdot (A + C)\).

The distributive law allows simplifying complex logical expressions into more manageable parts, which can be essential for reducing logic circuits and making computations more efficient. Recognizing when and how to apply the distributive property can lead to simplified, economically designed Boolean expressions in both algebraic and real-world settings.