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| Strategy | When to Use | Example |
|---|---|---|
| Direct Proof | Show P → Q directly | "If n is even, n² is even" |
| Contradiction | Assume ¬Q, derive contradiction | Proving √2 is irrational |
| Contrapositive | Prove ¬Q → ¬P instead | Logical equivalence |
| Induction | Statements about all n ∈ ℕ | Sum formulas |
| Cases | Different scenarios | Piecewise functions |
Claim: P(n) is true for all n ≥ 1
Base Case: Show P(1) is true.
[Verify for n = 1]
Inductive Step:
Assume P(k) is true for some k ≥ 1. (Inductive Hypothesis)
Show P(k+1) is true.
[Derive P(k+1) using P(k)]
Therefore, by induction, P(n) is true for all n ≥ 1. ∎
| Symbol | Meaning |
|---|---|
| ∀ | For all |
| ∃ | There exists |
| ∈ | Element of |
| ⊂ | Proper subset |
| ⊆ | Subset or equal |
| ∪ | Union |
| ∩ | Intersection |
| ℕ | Natural numbers {1,2,3,...} |
| ℤ | Integers {...,-1,0,1,...} |
| ℚ | Rational numbers |
| ℝ | Real numbers |
| ℂ | Complex numbers |
| ∞ | Infinity |
| ∴ | Therefore |
| ∵ | Because |
| ∎ | QED (proof complete) |