Monday, September 05, 2005
Methods of Proof
DIRECT PROOF:
At any stage of a proof, an implication may be replaced by its contrapositive {e.g., p = ~(~p)}. At any stage of a proof, any true expression may be replaced by another true expression {e.g., if the term sin²x is in an expression, (1 - cos²x) can be substituted}.
INDIRECT PROOF:
If ~p is false, then p is true. Therefore, to prove p is true, assume ~p is true and show that a direct proof with this assumption leads to a contradiction.
PROOF BY COINCIDENCE:
Assume two points or lines are different. Show that a direct proof with this assumption leads to a contradiction.
MATHEMATICAL INDUCTION:
If a set A contains all integers n≥a and for all n in A a proposition pn is to be proved, this can be done as follows:
(1) prove pa is true (the initial case);
(2) assume then that pk is true and prove pk--->pk+1 is true.
REVERSING STEPS:
Assume what is to be proved is already true. Perform valid operations until a result is obtained which is known to be true. Then reverse the steps (if this is possible) and the proof follows!
At any stage of a proof, an implication may be replaced by its contrapositive {e.g., p = ~(~p)}. At any stage of a proof, any true expression may be replaced by another true expression {e.g., if the term sin²x is in an expression, (1 - cos²x) can be substituted}.
INDIRECT PROOF:
If ~p is false, then p is true. Therefore, to prove p is true, assume ~p is true and show that a direct proof with this assumption leads to a contradiction.
PROOF BY COINCIDENCE:
Assume two points or lines are different. Show that a direct proof with this assumption leads to a contradiction.
MATHEMATICAL INDUCTION:
If a set A contains all integers n≥a and for all n in A a proposition pn is to be proved, this can be done as follows:
(1) prove pa is true (the initial case);
(2) assume then that pk is true and prove pk--->pk+1 is true.
REVERSING STEPS:
Assume what is to be proved is already true. Perform valid operations until a result is obtained which is known to be true. Then reverse the steps (if this is possible) and the proof follows!
