Part 02- Properties of Proposition law of excluded middle and law of contradiction.

Ali Seron Jason

6 days ago

https://www.gyanodhan.com/video/7B5.%20GATE%20CSEIT/Discrete%20Mathematics%202/524.%20Day%2004%20%20%20Part%2002-%20Properties%20of%20Proposition%20law%20of%20excluded%20middle%20and%20law%20of%20contradiction.mp4

Here is Part 02 of Discrete Mathematics – Properties of Proposition, focusing on two fundamental logical laws: the Law of Excluded Middle and the Law of Contradiction. These are essential for understanding propositional logic, used in mathematics, computer science, and GATE CSE.

Contents

1 What is a Proposition?
2 Law of Excluded Middle (LEM)
3 Law of Contradiction
4 Truth Table Comparison
5 Summary
6 Tip for GATE & CS Students:
- 6.1 Part 02- Properties of Proposition law of excluded middle and law of contradiction.
- 6.2 The Principle Of Excluded Middle Then And Now: Aristotle …

What is a Proposition?

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

Example:

“2 + 2 = 4” (True)
“It will rain today” (Not a proposition unless it has a definite truth value)

Law of Excluded Middle (LEM)

Definition:

For any proposition P:

$\lor \neg P \text{ is always TRUE}$

This means that either the proposition is true, or its negation is true. There’s no third option.

Example:

Let P = “5 is an odd number”

$\text{True}$
$¬P=False\neg P = \text{False}$
$\lor \neg P = \text{True} \lor \text{False} = \text{True}$

So the law holds.

Why It Matters:

Forms the basis of classical logic.
Used in proof by contradiction.
Helps in simplifying logical expressions in computer science.

Law of Contradiction

Definition:

For any proposition P:

$\land \neg P \text{ is always FALSE}$

This means a proposition cannot be both true and false at the same time.

Example:

Let P = “The Earth is flat”

$\text{False}$
$¬P=True\neg P = \text{True}$
$\land \neg P = \text{False} \land \text{True} = \text{False}$

So the law holds.

Why It Matters:

Ensures logical consistency
Used to reject contradictory systems
Essential in automated theorem proving and program correctness

Truth Table Comparison

P	¬P	P ∨ ¬P (LEM)	P ∧ ¬P (Contradiction)
T	F	T	F
F	T	T	F

Summary

Property	Statement	Always	Usefulness
Law of Excluded Middle	$\lor \neg P$	TRUE	Binary logic, Proof by contradiction
Law of Contradiction	$\land \neg P$	FALSE	Consistency checking, Contradiction test

Tip for GATE & CS Students:

These laws are used heavily in logical gates, compiler design, algorithms, and automata theory.
Practice with truth tables and logical identities.

Would you like practice questions, truth table exercises, or a PDF notes version of this topic?

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9e0.png" alt="🧠" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> What is a Proposition?

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Law of Excluded Middle (LEM)

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4d8.png" alt="📘" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Definition:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/2705.png" alt="✅" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Example:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9e0.png" alt="🧠" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Why It Matters:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f538.png" alt="🔸" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Law of Contradiction

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4d8.png" alt="📘" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Definition:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/2705.png" alt="✅" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Example:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9e0.png" alt="🧠" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Why It Matters:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4ca.png" alt="📊" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Truth Table Comparison

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4cc.png" alt="📌" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Summary

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9e0.png" alt="🧠" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Tip for GATE & CS Students: