techniques in mathematical proofs
The following example (from ring theory) illustrates the one aspect of proofs in mathematics: proving the existence of certain mathematical .
Statement: Let be a ring such that is right invertible, with . Then is right invertible.
This statement will be proven here using two methods. The first method is called an existential proof (also known as an existence proof), in which one only seeks to prove that the mathematical in question exists, not to show how to obtain it. The second method is called a constructive proof, in which one actually shows how to obtain the mathematical in question.
Existential proof: Since is right invertible, . Now,
showing that .
Notice, we merely demonstrated the existence of a right inverse of without actually finding such an . The next proof in fact finds a right inverse of .
Constructive proof: Since is right invertible, let be a right inverse so that . We seek to construct a right inverse of in terms of and . Rewriting the equation, we have . Then,
We have just expressed in terms of . Next, multiply on the right to each term on both sides of the equation, to get
Then, negate both terms and add 1, to get
Finally, rearranging the terms and we have
a right inverse of exists by explicitly constructing one.
Many other techniques are used in proving mathematical statements. Proof by mathematical induction, proof by contradiction, proof by contrapositive, and proof by exhaustion are just some of the major techniques (a is in the entry “irrational to an irrational power can be rational”).
As this entry is still in its very rough form, PM users are welcome and encouraged to refine and provide additional techniques with interesting and illustrative examples!
|Title||techniques in mathematical proofs|
|Date of creation||2013-03-22 14:46:15|
|Last modified on||2013-03-22 14:46:15|
|Last modified by||CWoo (3771)|