How do you write mathematical proofs? I don't understand.
I have a basic idea of how to write proofs, but even though I know this, I still can;t figure out how to write proofs for any statement. I just find it very confusing.
I have a basic idea of how to write proofs, but even though I know this, I still can;t figure out how to write proofs for any statement. I just find it very confusing.
1 Answer
Begin with given or known information. Apply a series of valid arguments. Conclude something new.
Explanation:
There is no simple formula for writing a proof, but the main idea is pretty constant. You begin with certain given information. You make valid arguments based off of this or other known information. These arguments eventually allow you to claim the conclusion.
There are many forms of proof. Students tend to be introduced to proofs through two-column proofs, in which statements are written in the left column, and their justifications in the right column:
or through flowchart proofs, in which statements are written in boxes, and the justification from moving from one statement to the next is written on arrows connecting them:
These are both forms of direct proof. There are many other types of proof. Examples of commonly used proof types include
-
Proof by contradiction: We assume that our conclusion is false, and then show that that must be false because it leads to a contradition.
-
Proof by contraposition: We use the fact that the truth of a proposition is equivalent to the truth of its contrapositive, and prove the contrapositive instead.
-
Proof by induction: We show that a proposition is true for some small integer, and show that its truth for one integer implies its truth for the next. We then claim that the proposition is true for all integers greater than our initial one using the property of mathematical induction.
-
Proof by cases: We divide up all possibilities into different cases, and then prove our claim for each one separately.
-
Proof by counterexample: We prove that a generalized claim is false by explicitly showing a case in which it is not true.
Given some arbitrary statement, it is not always obvious what technique to use, or how to proceed. There may be multiple ways to prove something, or no way at all within a given system. Courses are taught and textbooks written on understanding and applying different proof techniques.
However, again, the idea behind all of it is the same. Begin with what is given or known, and through valid arguments, conclude something new.
