Method of proof data structure

Reductio ad absurdum

• The basic concept:
  In general, assuming that the original proposition does not hold (that is, under the conditions of the original proposition, the conclusion does not hold), through correct reasoning, the conclusion that contradiction, therefore instructions assume that error, thus proving the original statement is true, such a proof method is called reductio ad absurdum .

• Basic operation:
  1. Distinguish propositions p => q conditions and conclusions;
  2. Q to make the proposition conclusion contradicts the assumed ┐q ;
  3. Starting from p and ┐q, applying the correct method of reasoning, the introduction of conflicting results ;

  4. The results concluded that the cause of conflict, is made by beginning assumed ┐q is not true , then the original conclusion q establishment , which indirectly proves the proposition p => q is true .

     The third step said conflicting results, generally refers to the introduction results with known conditions, definitions, theorems or temporarily assumed contradiction, and paradox and so on.

• Applicability:
  Applies to "the regular hard against" the proofs
  all "at least", "unique" or contain negative word proposition suitable by contradiction.

Mathematical induction

• The basic concept:
  from the initial situation, the gradual introduction of delivery n conclusion.
• Basic operation:
  Generally, demonstrated a positive integer n relating to the proposition, performed according to the following steps:
  1. (Induction foundation) demonstrated when n takes a first value \ (N - 0 \) when the statement is true;
  2. (Recursive induction) Suppose n-K = (k ≧ \ (N - 0 \) ) when the statement is true, demonstrated when n = k + 1 when the proposition is also true.
     As long as two steps above, you can conclude that (the introduction of delivery) proposition from \ (n_0 \) for all positive integers n have set the start.
• Note:
  When the proposition proved by mathematical induction, should pay attention to:
  1. The first step is the foundation, we must first verify = the n- \ (n_0 \) was established, the note \ (n_0 \) is not necessarily 1;
  2. The second step is the basis, in the second step, the key is to correct and reasonable use of induction hypothesis, in particular, to clarify the changes k to k + 1, the two steps are indispensable, and the writing must be standardized.
• Applicability:
  apply only to the positive integer n relating to proof of Proposition
• Added:
  mathematical induction can also solve the problem summarized guess the number of columns, the basic steps are: observation, induction, conjecture, proved in accordance with generally known conditions and recurrence relations, first find the first few series, and then summed summarized the law of which, guess conclusion, re-use mathematical induction to prove. I guess is proof of the premise and objects, so be sure to maintain the correctness of conjecture, while paying attention to the steps of writing mathematical induction.