Day 04Part 15(B) – Discrete Mathematics – Methods of solving Statements and boolean expressions.

Discrete Mathematics – Methods of Solving Statements and Boolean Expressions

In Discrete Mathematics, logical statements and Boolean expressions are fundamental concepts used in propositional logic, set theory, and digital logic design. Let’s explore different methods to solve them.

1. Logical Statements and Boolean Expressions

A statement (proposition) is a sentence that is either true (T) or false (F) but not both.

Boolean Variables & Operations

Boolean expressions use logical operators:

• AND ( ∧ ) → True if both inputs are true.
• OR ( ∨ ) → True if at least one input is true.
• NOT ( ¬ ) → Reverses the truth value.
• IMPLICATION ( → ) → If P → Q, Q must be true when P is true.
• BICONDITIONAL ( ↔ ) → True if both sides have the same truth value.

2. Methods of Solving Logical Statements

A. Truth Table Method

A truth table evaluates all possible truth values for a Boolean expression.

Example: Solve (P ∨ Q) → R

P	Q	R	P ∨ Q	(P ∨ Q) → R
T	T	T	T	T
T	T	F	T	F
T	F	T	T	T
T	F	F	T	F
F	T	T	T	T
F	T	F	T	F
F	F	T	F	T
F	F	F	F	T

Result: The expression is not a tautology (it is false in some cases).

B. Algebraic Simplification (Boolean Algebra)

Using Boolean algebra rules, expressions can be simplified.

Example: Simplify A + A’B

Solution:

1. Apply Absorption Law:
   A + A’B = (A + B)

Example: Simplify (A + B)(A + C)

Solution:

1. Apply Distribution Law:
   (A + B)(A + C) = A + BC

C. Karnaugh Map (K-Map) Method

Used for minimizing Boolean expressions, especially in digital logic.

Example: Simplify F(A, B, C) = Σ(1, 3, 4, 6, 7) using a K-map.

1. Draw a 3-variable K-map.
2. Group 1s into power-of-two blocks.
3. Derive the simplified equation.

3. Practical Applications

Digital Circuit Design – Simplifying Boolean expressions reduces hardware cost.
Computer Programming – Used in conditional statements and algorithms.
Mathematical Logic – Helps in theorem proving and automated reasoning.

Would you like more examples or a step-by-step K-map solution?

Day 04Part 15(B) – Discrete Mathematics – Methods of solving Statements and boolean expressions.

DIGITAL NOTES ON Discrete Mathematics B.TECH II YEAR

Discrete Mathematics for Computer Science

dmoi-tablet.pdf – Discrete Mathematics

Notes on Discrete Mathematics

DISCRETE MATHEMATICS

Here’s a structured lesson for:

Day 04 Part 15(B) – Discrete Mathematics

Methods of Solving Statements and Boolean Expressions

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PART A: Solving Logical Statements

1. Basic Logical Connectives

Symbol	Meaning	Example
¬P	NOT P	If P = true, then ¬P = false
P ∧ Q	P AND Q	True only if both are true
P ∨ Q	P OR Q	True if at least one is true
P → Q	Implication	“If P then Q”
P ↔ Q	Biconditional	True when P and Q have same truth value

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2. Truth Table Method

Example:

Statement: P→QP \rightarrow Q

P	Q	P → Q
T	T	T
T	F	F
F	T	T
F	F	T

Use this to prove tautology, contradiction, contingency.

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3. Logical Equivalence

Use laws to simplify:

Law	Example
De Morgan’s Laws	¬(P ∧ Q) ≡ ¬P ∨ ¬Q
Double Negation	¬(¬P) ≡ P
Implication	P → Q ≡ ¬P ∨ Q
Contrapositive	P → Q ≡ ¬Q → ¬P
Distributive Law	P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)

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PART B: Solving Boolean Expressions

1. Boolean Algebra Basics

Operation	Symbol	Rule
AND	·	A · B
OR	+	A + B
NOT	¬ or ′	¬A or A′

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2. Boolean Laws

Law	Expression
Identity	A + 0 = A, A · 1 = A
Null	A + 1 = 1, A · 0 = 0
Idempotent	A + A = A, A · A = A
Complement	A + A′ = 1, A · A′ = 0
Double Negation	(A′)′ = A

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3. Simplifying Boolean Expressions

Example:

Expression:
A + A·B

Apply Absorption Law:
A + A·B = A

Another:

Expression:
A·B + A·B′

Apply Distributive Law:
A·(B + B′) = A·1 = A

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Practice Problem Set

1. Prove using truth tables:
   • (P∧Q)→P(P ∧ Q) → P is a tautology.
2. Simplify Boolean expression:
   • A⋅(B+A′)A·(B + A′)
3. Check if:
   • P∨(¬P∧Q)≡P∨QP ∨ (¬P ∧ Q) ≡ P ∨ Q

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Quick Tips

• Use truth tables for 2–3 variable expressions.
• Use Boolean algebra rules to reduce expressions in logic circuits.
• Always look to factor, distribute, or absorb to simplify.

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Would you like:

• A PDF of Boolean Laws Cheat Sheet?
• Truth Table Generator (Python or Excel)?
• Or Practice Quiz with Solutions?

Let me know!

Day 04Part 15(B) – Discrete Mathematics – Methods of solving Statements and boolean expressions.

dmoi4.pdf – Discrete Mathematics – An Open Introduction

Discrete Mathematics – (Proof Techniques)