proofy.
Streak 00
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01one

Statements and logic

What counts as a statement, and how if-then works.

3 lessons0 done
  1. Lesson 001 · Locked · Read and choose
    What is a statement?
    A statement is a sentence that is definitely true or definitely false.
    Continue
  2. Lesson 002 · Locked · Read and choose
    If … then …
    When is “PP implies QQ” true?
  3. Lesson 003 · Locked · Read and choose
    Reading ∀ and ∃
    “For all” and “there exists”.
02two

Truth tables

Decide when compound statements are true, and when two statements say the same thing.

5 lessons0 done
  1. Lesson 004 · Locked · Read and choose
    Truth tables
    List every case and see what a compound statement does.
  2. Lesson 005 · Locked · Read, fill, write
    Equivalent statements
    Two statements are equivalent when their truth tables match.
  3. Lesson 006 · Locked · Read and write
    Logic gates
    Boolean algebra for circuits and code.
  4. Lesson 007 · Locked · Read, fill, write
    Negating an implication
    The negation of “if PP then QQ” is “PP and not QQ”.
  5. Lesson 008 · Locked · Decide
    Logic drill
    Eight quick decisions about truth and equivalence.
03three

Direct proof

Assume what is given, deduce step by step, and arrive at what you wanted to show.

4 lessons0 done
  1. Lesson 009 · Locked · Fill the gaps
    Definitions as tools
    Turn a word like “even” into an equation you can use.
  2. Lesson 010 · Locked · Put steps in order
    Assume, then deduce
    The shape of every direct proof.
  3. Lesson 011 · Locked · Fill, order, write
    Even plus even
    Your first full proof.
  4. Lesson 012 · Locked · Decide and write
    Direct proof drill
    Practice assuming, unpacking and concluding.
04four

Contrapositive

Prove P ⇒ Q by proving ¬Q ⇒ ¬P.

3 lessons0 done
  1. Lesson 013 · Locked · Read and choose
    The contrapositive
    Flip and negate: P⇒QP \Rightarrow Q becomes ¬Q⇒¬P\lnot Q \Rightarrow \lnot P.
  2. Lesson 014 · Locked · Fill and order
    Odd squares
    Prove “n2n^2 even ⇒n{}\Rightarrow n even” through its contrapositive.
  3. Lesson 015 · Locked · Decide and write
    Contrapositive drill
    Spot when the contrapositive is easier.
05five

Contradiction

Assume the opposite and reach an impossibility.

3 lessons0 done
  1. Lesson 016 · Locked · Read and choose
    Assume the opposite
    Suppose the claim is false and see what breaks.
  2. Lesson 017 · Locked · Fill and write
    No largest integer
    Your first proof by contradiction.
  3. Lesson 018 · Locked · Decide and write
    Contradiction drill
    Choose the right assumption, then reach the impossible.
06six

If and only if

Prove both directions, and prove that exactly one thing exists.

3 lessons0 done
  1. Lesson 019 · Locked · Read and order
    If and only if
    Prove P⇔QP \Leftrightarrow Q by proving both directions.
  2. Lesson 020 · Locked · Write the proof
    Both directions
    Prove nn is even exactly when 3n3n is even.
  3. Lesson 021 · Locked · Fill and write
    Exactly one
    Prove existence, then prove uniqueness.
07seven

Proof by cases

Split the problem into cases that cover everything.

4 lessons0 done
  1. Lesson 022 · Locked · Read and choose
    Splitting into cases
    Cover every possibility, one case at a time.
  2. Lesson 023 · Locked · Fill and order
    Parity cases
    Prove n2+nn^2 + n is always even.
  3. Lesson 024 · Locked · Write the proof
    Absolute value
    Prove x≤∣x∣x \le |x| by cases.
  4. Lesson 025 · Locked · Decide and write
    Cases drill
    Choose cases that cover everything.
08eight

Quantifiers

Prove and disprove statements with ∀ and ∃.

6 lessons0 done
  1. Lesson 026 · Locked · Read and choose
    Proving ∃ and ∀
    One example for ∃\exists. Every case for ∀\forall.
  2. Lesson 027 · Locked · Choose and fill
    Counterexamples
    Disprove a ∀\forall claim with one case.
  3. Lesson 028 · Locked · Read, choose, fill
    Negating quantified statements
    Flip each quantifier and negate what is left.
  4. Lesson 029 · Locked · Read and choose
    Order matters
    ∀∃\forall \exists and ∃∀\exists \forall say different things.
  5. Lesson 030 · Locked · Write the proof
    Proving ∀∃ claims
    Take an arbitrary xx, then build a yy from it.
  6. Lesson 031 · Locked · Decide
    Quantifier drill
    Mixed practice: prove, disprove, negate.
09nine

Writing it well

A proof written from a blank page.

1 lessons0 done
  1. Lesson 032 · Locked · Write from a blank page
    Boss proof: odd times odd
    Prove that the product of two odd integers is odd.