Logic & Propositions
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Master the foundations of mathematical reasoning: propositions, logical connectives, truth tables, and equivalences.
Concepts in this topic
- Propositions & Truth Values — Learn what propositions are and how truth values work
- Logical Connectives — Understand AND, OR, NOT, implication, and biconditional
- Truth Tables — Construct and analyze truth tables for compound propositions
- Logical Equivalences — Master key equivalence laws used to simplify logical expressions
Key formulas & identities
| Name | Formula | Meaning |
|---|---|---|
| De Morgan's Law (AND) | ¬(p ∧ q) ≡ ¬p ∨ ¬q | The negation of a conjunction is the disjunction of the negations |
| De Morgan's Law (OR) | ¬(p ∨ q) ≡ ¬p ∧ ¬q | The negation of a disjunction is the conjunction of the negations |
| Conditional Equivalence | p → q ≡ ¬p ∨ q | An implication is equivalent to the disjunction of the negated hypothesis and the conclusion |
| Contrapositive | p → q ≡ ¬q → ¬p | An implication is logically equivalent to its contrapositive |
| Biconditional Expansion | p ↔ q ≡ (p → q) ∧ (q → p) | A biconditional is the conjunction of both directions of implication |
| Distributive Law (AND over OR) | p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) | Conjunction distributes over disjunction |
| Distributive Law (OR over AND) | p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) | Disjunction distributes over conjunction |
| Absorption Law (1) | p ∨ (p ∧ q) ≡ p | A variable absorbs its conjunction with another |
Sample practice problem
From the free practice sample — 77 guided problems cover this topic in interactive practice.
Which of the following is a proposition?
- "What is your name?"
- "2 + 2 = 4"
- "x + 1 = 5"
- "Please sit down"
Show answer
Answer: "2 + 2 = 4"
"2 + 2 = 4" is the only declarative statement with a definite truth value (true). Questions, commands, and statements with variables are not propositions.