Algebra/Proofs

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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.

[edit] 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
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