Tautologies and Contradictions
Always true or always false statements.
Key idea
Always true or always false statements.
In depth
A tautology is true in every row (p ∨ ¬p); a contradiction is false in every row (p ∧ ¬p). Statements that are sometimes true and sometimes false are contingent. Tautologies are the laws of logic.
Worked example
(p ∧ q) → p is a tautology: if both are true, p is true.
Real-world uses
- Spotting always-true or never-true conditions in software code and logic puzzles
Common mistake
Calling any frequently true statement a tautology.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Tautologies and Contradictions?
Always true or always false statements. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Tautologies and Contradictions used in real life?
Real-world examples: Spotting always-true or never-true conditions in software code and logic puzzles.
What should I learn before Tautologies and Contradictions?
Make sure you are comfortable with Truth Tables, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Tautologies and Contradictions?
Calling any frequently true statement a tautology.
How can I practise Tautologies and Contradictions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Tautologies and Contradictions?
Move on to Logical Equivalence next.
Related advanced topics in Discrete Math & Number Theory
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.
- Proof by Direct Argument – Derive the conclusion from premises with valid steps.