In mathematics, a proof
is a series of logical steps of reasoning leading to a conclusion. The reasoning should be strict enough so that the conclusion must
be true if the given circumstances it uses are true. There are several different methods that can be used to construct a mathematical proof. Which method that is preferred in a specific case is highly dependent on the nature of the problem.
A direct proof
uses logical reasoning where one thing leads to next until the statement is shown to be true. One theorem that can be proven with a direct proof is the .
In a proof by contrapositive
, a claim is proven by stating and proving the statement. Because a statement and its contrapositive are equivalent, proving the first is the same as proving the second.
The former of these is the typical strategy used when creating a direct proof. The same statement can be proven by using a sequence of logical steps to show that not Q
leads to not P.
In a proof by mathematical induction
, one special case is shown to be true. Next, it is shown that if the statement is true for any special case, then it is also true for some other special case. This reasoning typically proves a statement to be true for an infinite number of special cases.
In a proof by contradiction
, a claim is proven by showing how the opposite conclusion of the claim creates a contradiction. A classic example is the proof that
is an irrational number. By assuming the opposite, that
is a rational number, it is possible to end up with a contradiction.
Traditionally, a proof ends with an abbreviation that tells you that the proof is concluded. A common example is Q. E. D. that comes from the Latin
Quod Erat Demonstrandum
, which means
what was to be shown