Quiz 2

Advanced Proof Techniques

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# Advanced Proof Techniques ## Proof by Cases **Example:** Prove $n^3 - n$ is divisible by 3 for all integers $n$. **Proof:** Consider $n = 3k$, $3k+1$, or $3k+2$ (all integers are one of these forms by division algorithm).

Advanced Proof Techniques

Proof by Cases

Example: Prove n3−nn^3 - n is divisible by 3 for all integers nn. Proof: Consider n=3kn = 3k, 3k+13k+1, or 3k+23k+2 (all integers are one of these forms by division algorithm).
  • n=3kn = 3k: n3−n=27k3−3k=3(9k3−k)n^3 - n = 27k^3 - 3k = 3(9k^3 - k) ✓
  • n=3k+1n = 3k+1: n3−n=(27k3+27k2+9k+1)−(3k+1)=27k3+27k2+6k=3(9k3+9k2+2k)n^3 - n = (27k^3 + 27k^2 + 9k + 1) - (3k+1) = 27k^3 + 27k^2 + 6k = 3(9k^3 + 9k^2 + 2k) ✓
  • n=3k+2n = 3k+2: n3−n=(27k3+54k2+36k+8)−(3k+2)=27k3+54k2+33k+6=3(9k3+18k2+11k+2)n^3 - n = (27k^3 + 54k^2 + 36k + 8) - (3k+2) = 27k^3 + 54k^2 + 33k + 6 = 3(9k^3 + 18k^2 + 11k + 2) ✓

Combinatorial Proofs

Prove ∑k=0n(nk)=2n\sum_{k=0}^n \binom{n}{k} = 2^n: Combinatorial argument: The RHS counts subsets of an nn-element set. The LHS sums the number of subsets of each size kk. Both count the same thing, so they're equal. Join Discord PreviousMathematical InductionNextSet Theory
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