that is sometimes challenging (or impossible) to prove the a conjecture is true using direct methods. For example, to show that the square source of 2 is irrational, us cannot straight test and reject the infinite number of rational numbers whose square might be two. Instead, we show that the presumption that root 2 is rational leader to a contradiction. The steps taken because that a proof by contradiction (also called indirect proof) are: i think the the contrary of her conclusion. for “the primes are limitless in number,” assume the the primes are a finite collection of size n. Come prove the declare “if a triangle is scalene, climate no 2 of its angles room congruent,” assume that at least two angles are congruent. use the assumption to derive brand-new consequences till one is the opposite of her premise. For the two instances above, friend would seek to establish: the there exists a prime not counted in the initial collection of n primes. That the triangle cannot be scalene. Conclude the the assumption must it is in false and also that its the contrary (your initial conclusion) must be true. Why walk this an approach make sense? One way to recognize it is to note that friend are producing a direct proof the the contrapositive of your original statement (you are proving if not B, then no A). Due to the fact that contrapositive statements are constantly logically equivalent, the initial then follows. note that the contradiction pressures us to reject our assumption because our other steps based on that presumption are logical and justified. The just “mistake”that we can have made was the assumption itself. An indirect proof develops that the contrary conclusion is not consistent with the premise and that, therefore, the original conclusion need to be true.

Opposites

With the republicans in power, it’s man eat man. V the Democrats, it’s just the opposite. — Bumper sticker.

Sometimes, it can be a difficulty determining what opposing of a conclusion is. The contrary of “all X are Y” is not “all X are not Y,” yet “at the very least one X is not Y.” Similarly, when we have a link conclusion, we should be careful.


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Think about these two examples:

The original Conjecture The opposite of the Conclusion
If m and also n are integers and mn is odd, then m is odd and n is odd. m is also or n is even
If m + n is irrational, then m is irrational or n is irrational. m is reasonable and n is rational

Resources

check out Triangle with limited Angle sum for a exercise problem and also Proof by Contradiction Class task for a lesson plan that introduces proof by contradiction.

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