Algebra/Proofs

An Algebraic Proof is a proof of a Theorem (or Lemma or Corollary) that relies highly on symbolic manipulation of equations to go from the assumption(s) to the conclusion(s). An algebraic proof may be contrasted with a combinatorial proof, which involves more discussion of the objects themselves, rather than the symbols representing them.

Example

Theorem: $(\binom{n}{k}) + (\binom{n}{k + 1}) = (\binom{n + 1}{k + 1})$ .

Proof:

  $L H S = \frac{n!}{k! (n - k)!} + \frac{n!}{(k + 1)! (n - k - 1)!}$ 
  $= \frac{n! (k + 1)}{(k + 1)! (n - k)!} + \frac{n! (n - k)}{(k + 1)! (n - k)!}$ 
  $= \frac{(k + 1 + n - k) n!}{(k + 1)! (n - k)!} = \frac{(n + 1) n!}{(k + 1)! (n - k)!} = R H S$

Algebra/Proofs

Example

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