Truth table builder
Type a statement using P, Q and R. Add a second one to test whether they are equivalent.
Tap a cell to cycle ? → T → F.
Keyboard: ~ or ! for ¬, ^ or & for ∧, v or | for ∨, => for ⇒, <=> for ⇔.
Station 01 · The language and notation of proof
Statements and logic
A statement is a sentence that is either true or false, but not both. Build truth tables to see when two statements say the same thing.
- statement / proposition
- P ∧ Q, P ∨ Q, ¬P
- P ⇒ Q, P ⇔ Q
- De Morgan's laws
- P ⇒ Q ≡ ¬P ∨ Q
- equivalence
Is it a statement?
A statement must be either true or false, but not both. It does not matter whether we know which.
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The connectives
Station 02 · Implication, converse, contrapositive, negation
One implication, four relatives
An implication and its contrapositive always have the same truth value. The converse may or may not be true. Decide each, then reveal.
- implication P ⇒ Q
- converse Q ⇒ P
- contrapositive ¬Q ⇒ ¬P
- negation P ∧ ¬Q
- counterexample
- ∀ and ∃
Decide true or false
0 / 0Negating quantifiers
To negate, swap ∀ and ∃ and negate what follows. Choose the correct negation.
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Why the contrapositive works
Station 03 · Illustrations of proofs
Proof by contradiction and contrapositive
Put the lines of each proof in order. Watch for decoy lines that look plausible but are wrong or unnecessary.
- proof by contradiction
- proof via the contrapositive
- results involving integers
Contradiction: the template
Contrapositive: the template
Station 04 · Examples, counterexamples and integers
Counterexamples and proof by cases
One counterexample is enough to disprove a "for all" claim. No number of examples proves one. To prove results about integers, split them into cases by remainder.
- examples and counterexamples
- results involving integers
- proof by cases
Counterexample hunter
Divisibility by cases
Every integer is dk + r for some remainder r. Check what the expression leaves in each case.
Station 05 · Proof of inequalities
Proving inequalities
Use a > b ⇔ a − b > 0, the fact that squares are non-negative, known inequalities, geometry, graphs and calculus.
- a > b ⇔ a − b > 0
- (a ± b)² ≥ 0
- rules for inequalities
- triangle inequality
- AM ≥ GM
- squeeze theorem
- graphical and calculus methods
Rule or myth?
Move the sliders to hunt for counterexamples. A card turns red when its "if" part holds but its "then" part fails. Then decide whether each rule is always true.
AM ≥ GM, geometrically
The diameter is a + b. The radius is the arithmetic mean; the half-chord at the join is the geometric mean.
The triangle inequality |a + b| ≤ |a| + |b|
Distances on the number line: walking a then b never ends further from 0 than |a| + |b|.
Squeeze, graphs and calculus
Station 06 · Further proof by mathematical induction
Mathematical induction
Prove the first case, then prove that each case forces the next. Both parts are essential.
- identities and the binomial theorem
- divisibility
- inequalities
- calculus
- recursive formulas
- geometric results
The dominoes
Tick what has been proved, then knock them over.
Conjecture first
Recurrence tester
Define a sequence recursively, conjecture a formula, and see whether they agree before you prove it.
Use a for aₙ and n for n. Try a₁ = 2, aₙ₊₁ = a + 2n + 2, aₙ = n(n+1).
Writing the inductive step
Station 07 · Critical reading
Spot the error
Each "proof" reaches a false or unjustified conclusion. Tap the first line that does not follow.
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Common traps
Station 08 · Practice
Quick-fire practice
Random questions across the topic, each with a worked explanation.
Explanation
Choose an answer to see the explanation.