1, distinguish the title and conclusion of the proposition "If A is B";
2. Make assumptions that contradict proposition conclusion B;
3. Starting from A sum, applying the correct reasoning method, we get contradictory results;
4. The reason for the contradictory results is that the assumptions made at the beginning are incorrect, so the original conclusion B is established, which indirectly proves the original proposition. Generally speaking, in the proof, we should pay attention to the rigor of reasoning, have evidence step by step, and really understand where the contradiction lies.
For example:
Two straight lines intersect and there is only one intersection.
It is known that A and B are two intersecting straight lines.
Prove that there is only one intersection between A and B.
Prove:
For example, two straight lines intersect and there is only one intersection.
It is known that A and B are two intersecting straight lines.
Prove that there is only one intersection between A and B.
Prove:
Assuming that lines A and B have more than one intersection point, then they have at least two points intersecting.
Let these two intersections be e and f.
Then, line A passes through points E and F, and line B also passes through points E and F..
In other words, two straight lines A and B can be made after two points E and F..
This contradicts the axiom that a straight line can be made after two points, and only one straight line can be made.
So the assumption doesn't hold.
Then the original problem "two straight lines intersect with only one intersection point" holds.
Extended data applicable to reduction to absurdity 1:
In the process of proving some propositions, it is difficult to get proof directly from the original questions, and some even can't find the basis for proof in specific occasions. When it is difficult to draw a conclusion directly from the known conditions, reduction to absurdity can be considered.
2. Deduce the main contradiction types in the process of reasoning by reduction to absurdity.
The result of (1) is inconsistent with the known axiom;
(2) The result of deduction contradicts the known theorem;
(3) The inferred result contradicts the hypothesis;
(4) The deduced result is inconsistent with the known definition;
(5) Two contradictory results are deduced;
(6) The deduced result contradicts the known conditions;
3. Problems that should be paid attention to when applying reduction to absurdity.
(1), because it is assumed that the conclusion of the proposition to be proved is not valid, all possible situations opposite to the conclusion must be considered. If there is only one, then it is ok to deny this; If there are many contradictions with the original proposition, we must deny them one by one, and remember not to miss anything, otherwise the original proposition will be difficult to prove.
(2) The reasoning process must be completely correct, otherwise, even if the contradictory results are deduced, it is not enough to prove that the hypothesis is wrong.
(3), in the process of reasoning, must use the known conditions, otherwise either can't deduce contradictory results, or can't conclude that the result is wrong.