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: $$ {n \choose k} + {n \choose k+1} = {n+1 \choose k+1}$$.

Proof: $$ LHS = \frac{n!}{k!(n-k)!} + \frac{n!}{(k+1)!(n-k-1)!} $$ $$= \frac{n!(k+1)}{(k+1)!(n-k)!} + \frac{n!(n-k)}{(k+1)!(n-k)!}$$ $$= \frac{(k+1+n-k)n!}{(k+1)!(n-k)!} = \frac{(n+1)n!}{(k+1)!(n-k)!}=RHS$$