If fortune cookie lucky numbers worked,
then fortune cookies would be banned in Las Vegas.
But they are not banned in Las Vegas.
Ergo, the lucky numbers do not work.

If paleolithic diets led to long healthy lives,
then hunter-gatherers would have long life expectancies.
But hunter-gatherers would have short life expectancies.
Ergo, paleolithic diets do not lead to long healthy lives.


Modus Tolens
      A → B
      ¬B
    ∴ ¬A








Proof by contradiction
    To prove ¬A,
        Assume A
            Prove A → B
            Prove ¬B
        Conclude ¬A

Another way:
      A → ⊥
    ∴ ¬A
because
      A → ⊥
    ⊨ A → (B → ¬B)
    ⊨ (A ∧ B) → ¬B
    ⊨ (A → B) → ¬B






No one opened the door.
Proof:
    Assume someone had opened the door.
    Then the alarm would have sounded.
    But the alarm did not sound.
    That's a contradiction.
    Thus, the assumption that someone came in has to be false.










∄ q ∈ Q
    Assume ∃ q ∈ Q.
    Let q0 be one such element of Q.
    Because it's an element of Q, ...
    ...
    ... which means P(q0) is true.
    But P(q0) → q0 ∉ Q.
    This is a contradiction.
    Thus, ¬(∃ q ∈ Q),
    meaning ∄ q ∈ Q


Q = Set of people sitting in two chairs in this room
    Assume Q is not empty
    let q0 be an element of Q
    Because q0 is sitting in two chairs,
    and because there are arm rests or backs of chairs between chairs,
    and because arm rests and backs of chairs are higher than seats
    then q0 is sitting higher than everyone else.
    But I'm looking at the room, and no one is sitting higher.
    So q0 does not exist.
    But we assumed q0 did exist
    which is a contradiction
    ∴ Q is empty





Theorem:
    |N| < |R|
Proof
    assume |N| = |R|
    then ∃ f:N ←→ R (i.e., f is an invertible total function)
    consider one such f, f'.
    Define a number x as follows:
        all digits left of the decimal point in x are 0
        the digit in the 10^(-n) place is
            (f'(n) + 1) mod 10
    x is not in the range of f