A Brief Guide to Writing Beautiful Mathematical Proofs
A mathematical proof is a sequence of logical statements, each one being either an assumption
or a conclusion clearly following from an assumption or previously proved result. A proof
should flow naturally; you should be able to read it aloud in complete sentences.
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General Guidelines
The following list is from (1).
• A good proof should lead the reader through your logical arguments, with no assumption that the reader already knows the solution. Using correct English grammar and
punctuation will make your logical arguments easier to comprehend for the person
reading them.
• Always write down the statement you are going to prove, including any assumptions,
as your first line in the proof. Clearly indicate where the proof begins (e.g., Pf:) and
where it ends (e.g., QED 1 or ).
• Keep your reader informed. Indicate when you use assumptions and justify each step
of your argument.
• Use mathematical symbols correctly. All mathematical symbols and statements have
an English translation; you can check whether they make sense by reading your work
back to yourself.
Some common mathematical symbols are:
Symbol
=⇒
Read As
implies; if. . . then
⇐⇒
if and only if
Explanation
A =⇒ B means if A is true, then B is
also true; if A is false, then nothing is said
about B.
A ⇐⇒ B means A =⇒ B and B =⇒
A.
∀
∃
−
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−, s.t., :, or |
for all; for any; for each
there exists
such that
∈ or ∈
/
⊂ or 6⊂
∨
is or is not an element of
is or is not a subset of
or
∧
and
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Example: limx→a f (x) = L if ∀ > 0, ∃
δ > 0 −
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− |f (x) − L| < whenever 0 <
|x − a| < δ.
x ∈ V means x is an element of the set V .
W ⊂ V means W is a subset of the set V .
The statement A ∨ B is true if A or B (or
both) are true; if both are false, the statement is false.
The statement A ∧ B is true if A and B are
both true; else it is false.
Q.E.D. are the initials of the Latin phrase quod erat demonstrandum, which translates as “which was to
be demonstrated.”
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Methods of Proof
Part of the following list is from (3).
Direct Proof: To prove an implication A =⇒ B, assume A and derive B.
Proof by Cases: To prove (A or B) =⇒ C, prove A =⇒ C and B =⇒ C.
Proving “and” statements: To prove A =⇒ (B and C), prove A =⇒ B and A =⇒ C.
Proof by Contrapositive: To prove A =⇒ B, it is equivalent to prove the contrapositive
(not B) =⇒ (not A).
Proof by Contradiction: To prove A, assume not A and prove any contraction (e.g., B
and not B).
Proving “or” statements: To prove A =⇒ (B or C), procede by contradiction. Assume
A, not B and not C, then derive a contradiction.
Proof of “if and only if” statements: To prove A ⇐⇒ B, prove A =⇒ B and
B =⇒ A.
Proofs involving quantifiers: To prove A ∀ x ∈ V , you must show A is true whatever
value c is substituted for x. However, to disprove A ∀ x ∈ V , you only need to find one
example x = d such that A is false.
To prove ∃ x ∈ V −
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− A, you just need to find one example x = c for which A is true. To
disprove ∃ x ∈ V −
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− A, you need to show A is false for every possible value of x ∈ V .
Proof by induction: When proving a statement A for n = 1, 2, 3, . . . ,, proof by induction
involves two steps: an initial step and an inductive step. First, verify that A is true for
n = 1. Next, assume A is true for n = k and show A is true for n = k + 1. (Note: In general,
n can have any starting point. If you want to prove A for n = m, m + 1, m + 2, . . ., where
m is an integer, the initial step is to verify A for n = m. Then follow with the inductive step.)
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References
(1) A Brief Guide to Writing Proofs, NYU Polytechnic University
http://www.math.poly.edu/courses/ma2312/WritingProofs.pdf
(2) How to Write Proofs, Larry W. Cusick, California State University, Fresno
http://zimmer.csufresno.edu/∼larryc/proofs/proofs.html
(3) Basic Proof Methods, David Marker, University of Illinois at Chicago
http://www.math.uic.edu/∼marker/math215/methods.pdf
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