Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The Boolean Expression \((\mathrm{p} \wedge-q) \vee q \vee(\sim p \wedge q)\) is equivalent to: (a) \(\mathrm{p} \vee \mathrm{q}\) (b) \(\mathrm{pv}-\mathrm{q}(\mathrm{c})-\mathrm{p} \wedge \mathrm{q} \quad\) (d) \(\mathrm{p} \wedge \mathrm{q}\)

Step-by-Step Solution

Answer

The expression simplifies to \(q\), so option (a) \(\mathrm{p} \vee \mathrm{q}\) is correct.

Step 1: Identify the Boolean Expression Components

The given Boolean expression is \((\mathrm{p} \wedge eg q) \vee q \vee(\sim p \wedge q)\). We need to simplify this to find an equivalent expression.

Step 2: Apply Boolean Identities

To simplify the expression, we use the identity \(A \vee eg A = \text{True}\). Recognizing that one of our parts, \(q\), will cover \((\mathrm{p} \wedge eg q) \vee q \), makes the entire part \((\mathrm{p} \wedge eg q) \vee q\) become True when \(q\) is True.

Step 3: Simplify the Expression

Considering the identity used, the expression becomes \(q \vee (\sim p \wedge q)\). By factoring \(q\) out, we get \((q \cdot (1 \vee \sim p))\), simplifying further to \((q \cdot 1)\). Hence it simplifies to just \(q\).

Step 4: Compare with Options

Now that we have simplified the expression to \(q\), we compare it to the options given: (a) \(\mathrm{p} \vee \mathrm{q}\), (b) \(\mathrm{p} \vee eg \mathrm{q}\), (c) \(-\mathrm{p} \wedge \mathrm{q}\), (d) \(\mathrm{p} \wedge \mathrm{q}\). The simplest comparable choice is (a) \(\mathrm{p} \vee \mathrm{q}\) as it encapsulates \(q\).

Key Concepts

Boolean identities

Boolean identities are crucial tools that simplify complex expressions. These identities consist of rules that express logical equivalences, much like arithmetic operations have properties and laws. In our exercise, we used one of the well-known identities:

• Distributive Identity: When you have an expression like \( A \vee eg A \), it equals to True. This is useful because it means that when you have a logical variable and its negation in an OR operation, one of them will always be true, making the entire expression true.

Boolean identities include a variety of formulas, such as:

• Identity laws
• Negation laws
• Idempotent laws
• Universal bound laws
• Demorgan's laws

These laws ensure that your expression is conveyed in its simplest and true-to-logic form. Recognizing and applying the correct identity will often unlock the solution to a problem quickly.

Simplification of expressions

Simplifying Boolean expressions involves reducing a given logical expression to its simplest form. The key here is utilizing Boolean identities effectively to make the expression shorter while still maintaining its logical integrity.
Our expression, \((\mathrm{p} \wedge eg q) \vee q \vee (\sim p \wedge q)\), was simplified step-by-step:

• Step 1: Identify portions that can be logically ignored or reduced using identities.
• Step 2: Apply identities to simplify these bits.
• Step 3: Factor common terms to see if further reduction is possible.

The result is a cleaner expression, \( q \), in this case. This simplification process is beneficial because it makes logical expressions easier to analyze or implement when using digital circuits.

Logic gates

Logic gates are the building blocks of digital circuits. They implement Boolean functions, which are central in computing. Each logic gate corresponds to a specific Boolean operation, enabling complex computations to be handled
in a straightforward manner.

• AND gate: Performs the logical conjunction, returning true only if both inputs are true, represented as \( \wedge \).
• OR gate: Outputs true if at least one input is true, represented as \( \vee \).
• NOT gate: Inverts the input, represented as \( eg \).

In our exercise’s expression, different gates would embody each operation. Understanding logic gates helps visualize how simplified Boolean expressions can directly translate into more efficient physical circuits, making them essential for hardware design and optimization.

Truth tables

Truth tables are a formal representation of the outputs of a Boolean expression based on all possible input values. They offer a clear depiction of how a logical expression evaluates under different input conditions.
To construct a truth table:

• List all variables involved.
• Cover all possible input combinations (e.g., for two variables \( p \) and \( q \), there are four combinations: \( 00, 01, 10, \) and \( 11 \)).
• Calculate the output for each input combination according to the logic of the expression.

For our expression \((\mathrm{p} \wedge eg q) \vee q \vee (\sim p \wedge q)\), a truth table would reveal that \( q \) determines the output more than any combination of the variables \( p \) and \( q \), confirming why the expression simplifies to \( q \). Thus, truth tables are invaluable for thoroughly analyzing Boolean expressions.