We've been doing formal proofs

Moving to informal proofs

• Every step isn't outlined
• Axioms (like algebraic laws) are implied
• Some steps are combined
• Some things (like quantifiers) are implied

Most of the proofs we'll be going over are of the form 

∀x P(x) → Q(x)

How do we do this?

• Choose a proof strategy
• Use axioms, rules of inference, and previous results
• When proving ∀x P(x) → Q(x), we try to prove
  P(c) → Q(c)

Biconditional Proofs

• Prove ∀x P(x) ↔ Q(x)
• Similar to regular proofs, but need to prove P(c) → Q(c) and Q(c) → P(c)

Strategies for winning at proofs

Direct proofs

• Given a conditional statement p → q
• Prove if p is true, then q is also true
• Most direct type (duh), but may not lead you to a useful conclusion

Example

Prove that the product of two odd integers is odd.

Proof by Contraposition

• An "Indirect Proof"
• Instead of proving p → q, prove ¬q → ¬p

Example

Prove "If x² is even, then x is even"

Proof by contradiction

• Prove ¬p → (r ^ ¬r) for some proposition r
• In other words, assume that the premise is false but the conclusion is true. Then show that the conclusion is also false.

Example

If n is an integer and n^3 + 5 is odd, then n is even

Other types of proofs

• Vacuous Proof: Given p is false, p → q is always true
• Example: Prove P(0) is true where P(n) = "If n is a positive integer, then n^2 > n"
• Trivial Proof: Establish that q is true, therefore it doesn't matter what p is.
• Example: "If a and b are positive real numbers, then (a + b) ^n < (a^n + b^n)". Show that P(0) is true.

Proving Statements False

• Find a counterexample (i.e. p is true, but p → q is false)
• Example: Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.