Mathematical Logic - Interactive Simulator

Section 01

Introduction, Concept, and Statements

Step Discovery:

What is Logic?

Mathematical Logic is the study of principles and methods used to distinguish correct reasoning from incorrect reasoning. Formulated systemically by Aristotle and modernized by Leibniz, Boole, and Frege, it forms the foundation of modern mathematics, programming, and proof theory.

Key Insight:

In mathematics, logic allows us to rigorously prove theorems and construct clear, unambiguous arguments that hold universally true.

The Goal of Logic

To eliminate ambiguity and define truth value in logical formulations with mathematical precision.

Core Concepts in Logic

In general language, we make statements that are subjective, emotional, or conditional. In mathematical logic, we deal with formal mathematical systems. The basic elements we analyze are declarative sentences called propositions or statements.

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Objectivity

Sentences must have a clear truth value unaffected by personal opinion.

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Bivalence Law

A statement is either True (T) or False (F). It cannot be both or in-between.

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Formality

Focuses on form rather than subject content.

Definition of a Logical Statement

"A logical statement (or proposition) is a declarative sentence which is either strictly true or strictly false, but not both simultaneously."

Sentences that are interrogative (questions), imperative (commands/requests), exclamatory (emotions), or open (containing variables) are not statements in mathematical logic.

Statement or Not? Interactive Practice

Click each card to dynamically check if the sentence is a valid logical statement under the CDC curriculum guidelines!

Sentence 1

"The sum of 3 and 4 is 8."

Sentence 2

"Wow! What a beautiful day!"

Sentence 3

"Mt. Everest is in Nepal."

Sentence 4

"Open the window immediately."

Sentence 5

"x + 5 = 10"

Sentence 6

"All prime numbers are odd."

Section 02

Simple, Compound Statements and Truth Tables

Step Discovery:

01. Simple Statement

Atomic Propositions

A statement is said to be simple (or atomic) if it cannot be broken down further into simpler statements. It contains a single subject and predicate.

Standard Examples:

p: Kathmandu is the capital of Nepal.
q: 5 is a prime number.

02. Compound Statement

Molecular Propositions

A statement is called a compound (or molecular) statement if it is formed by combining two or more simple statements using specific logical connecting words (such as 'and', 'or', 'if...then', etc.).

Standard Examples:

p & q: Kathmandu is the capital of Nepal AND 5 is a prime number.

Concept of Truth Value

The truth or falseness of a logical statement is called its truth value.

True Statement

If a statement is facts-aligned, its truth value is T (or represented numerically as 1).

Example: "The Sun rises in the East." (T)

False Statement

If a statement contradicts factual truth, its truth value is F (or represented numerically as 0).

Example: "Pi (π) is a rational number." (F)

What is a Truth Table?

A Truth Table is a schematic, tabular visualization that lists all possible truth value configurations of component simple statements along with the corresponding resulting truth values of the compound statement.

Formula for Rows:

For $n$ independent statements, the truth table contains exactly 2ⁿ distinct rows of logical permutations.

Logical Permutation Table (For 2 statements, p and q)

p	q	Combination Description
T	T	Both components are True
T	F	First is True, second is False
F	T	First is False, second is True
F	F	Both components are False

Section 03

Logical Connectives & Truth Tables

Step Discovery:

Connective 1

Negation (NOT - ~p)

Negation swaps the truth value of a statement. If statement $p$ is True, then its negation $\sim p$ is False, and vice versa. It corresponds to the English word "NOT".

p	~p (Negation)
T	F
F	T

Connective 2

Conjunction (AND - p ∧ q)

A conjunction is True ONLY IF both constituent statements are True. If at least one statement is False, the entire conjunction is False.

p	q	p ∧ q (Conjunction)
T	T	T
T	F	F
F	T	F
F	F	F

Connective 3

Disjunction (OR - p ∨ q)

A disjunction is True if AT LEAST ONE of the statements is True. It is False only when both inputs are strictly False.

p	q	p ∨ q (Disjunction)
T	T	T
T	F	T
F	T	T
F	F	F

Connective 4

Conditional (If... Then - p ➔ q)

Also known as implication. It states that if $p$ occurs, then $q$ must occur. It is FALSE ONLY WHEN the hypothesis $p$ is True but the conclusion $q$ is False.

p (Hypothesis)	q (Conclusion)	p ➔ q (Implication)
T	T	T
T	F	F
F	T	T
F	F	T

Connective 5

Biconditional (If and Only If - p ↔ q)

A double implication ($p \leftrightarrow q$). It is TRUE ONLY WHEN both $p$ and $q$ share the exact same truth value (either both are True, or both are False).

p	q	p ↔ q (Equivalence)
T	T	T
T	F	F
F	T	F
F	F	T

Section 04

Tautology, Contradiction, Variations and Equivalence

Step Discovery:

Tautology (T)

A compound statement is a tautology if its truth value is ALWAYS True for all possible combinations of truth values of its simple components.

Classic Example: p ∨ ~p (Law of Excluded Middle)

Contradiction (C)

A compound statement is a contradiction (or absurdity) if its truth value is ALWAYS False under all logical truth assignments.

Classic Example: p ∧ ~p

Variations of Conditional Statement (p ➔ q)

From a given implication p ➔ q, we can formulate three other secondary implications:

01. Converse

q ➔ p

Interchanging hypothesis and conclusion.

02. Inverse

~p ➔ ~q

Negating both hypothesis and conclusion.

03. Contrapositive

~q ➔ ~p

Negating and interchanging. Mathematically equivalent to the original implication!

Logical Equivalence (≡)

Two compound statements are said to be logically equivalent if they yield identical truth values in every corresponding row of their truth tables.

Implication and Contrapositive Equivalence:
Let's verify why (p ➔ q) ≡ (~q ➔ ~p) using the systematic truth table.

p	q	~p	~q	p ➔ q	~q ➔ ~p
T	T	F	F	T	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

Section 05

Laws of Logic & Interactive Verification

Select a Logic Law to Investigate:

Tip: Use the selector above to verify different algebraic logical laws dynamically.

Section 06

Interactive Practice & Writing Workspace

Activate the Pen Tool below to write on-screen!

Practice Questions (CDC Class 11 Aligned)

Question 01

If p: "Nepal is a landlocked country" and q: "Kathmandu is in Asia", write the symbolic form and determine the truth value of the compound statement: "Nepal is a landlocked country and Kathmandu is not in Asia".

Question 02

Prove the logical equivalence relation of the Implication Statement and its Contrapositive, i.e., show (p ➔ q) ≡ (~q ➔ ~p) by constructing a complete truth table.

Question 03

Write the converse, inverse, and contrapositive statements for: "If x is divisible by 4, then x is divisible by 2."

Question 04

Define Tautology and Contradiction. Verify logically whether the proposition (p ➔ q) ↔ (~p ∨ q) is a tautology.

Question 05

State the Law of Syllogism and explain its real-world implementation in logical mathematics deduction proofs.

Question 06

Prove the logical equivalence relation of the Distributive Law: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) using truth table verification.

Local Digital Scratchpad / Writing Board

You can use this dedicated canvas pad to write down equations, draft tables, or calculate logical truth configurations with the Pen Tool activated.