MATH 13: FINAL EXAM NOTES
This isn’t meant to be a comprehensive study guide, but it provides some advice on studying for
our final exam.
Some topics from the course:
• Direct proof
• If-then statements (and other ways to write if-then statements such as using “only if” or
“implies”)
• Contrapositive
• Contradiction
• Logic (truth tables, etc.)
• Sets (especially set-builder notation. What is a finite set?)
• Proofs by cases
• Induction (and strong induction)
• Counter-examples
• Definitions
• Quantifiers
• Functions (domain, codomain, injective, surjective, bijective)
• Cardinality (when do two infinite sets have the same cardinality?)
• Relations and equivalence relations (most important: given a relation, is it reflexive? symmetric? transitive?)
Here are some practice definitions. For each, you should try finding some examples of sets/numbers/etc
that satisfy the definition to get a feeling for it.
1. A subset S ✓ R is called wide if for every x 2 R there exists s 2 S such that |x
s| < 2.
2. A subset S ✓ R is called narrow if there exist real numbers m, n 2 R such that for every s 2 S,
we have m < s < n.
3. A subset S ✓ Z>0 is called 3-primary if for all s1 , s2 , s3 2 S, we have that s1 + s2 + s3 is not
prime.
4. A subset U ✓ R2 is called divisible if there exists a vertical line satisfying the following three
properties:
• The vertical line does not intersect U ;
• There exists u 2 U to the left of the vertical line;
• There exists u 2 U to the right of the vertical line.
5. An integer n 2 Z>0 is called 3-composite if it is divisble by exactly three distinct primes.
6. A subset T ✓ Z is called flipped if for all x, y 2 T , we have xy 62 T . (Notice that when we say
“for all x, y 2 T ”, this includes the possibility x = y.)
There is no perfect strategy for proving theorems, but here is some general advice.
• Try the simplest examples you can.
a. Prove or disprove: Every set S ✓ R2 is star-shaped.
b. Give examples of sets satisfying/not satisfying each of the above definitions.
c. Prove or disprove: If n 2 Z, then n2 + 4n + 1 is even.
• If you try proving something and your proof fails, maybe that can give you an idea for a
counter-example.
a. Prove or disprove: If f (x) : R ! R is strictly increasing, then g(x) = (f (x))2 is strictly
increasing.
b. Prove or disprove: If f (x) : R ! R is given by f (x) = mx + b, then f (x) is 1-1.
• You will earn 1 bonus point for each of the following that you do:
a. Write your proofs always using complete sentences.
b. Avoid using the symbols ), !, 8, 9 within your proofs. Write them out using words
instead.
c. Do not start a sentence of a proof with a variable name.
• You will lose 2 points if you state something you are trying to prove without clearly indicating it is what you’re trying to prove. For example, say you want to prove that if n2 is
even, then n is even.
a. Good first sentence: “We will prove the contrapositive: If n is odd, then n2 is odd.”
b. Bad first sentence: “Proof by contrapositive: if n is odd, then n2 is odd.”
• Do not use a variable without saying what it is. For example:
a. Good: “By definition of odd, we know n = 2k + 1 for some k 2 Z.”
b. Bad: “By definition of odd, we know n = 2k + 1.”
• For tips on mathematical writing, see Section 5.3 of Hammack’s Book of Proof.
• If you’re disproving a statement, try to use as specific a counter-example as possible. I
prefer the statement “it is false for n = 1” to the statement “it is false for all negative
numbers”.
a. Prove or disprove: The function f (x) : R ! R given by f (x) = x2 is periodic with
period 2⇡.
• I don’t think it’s possible to disprove a statement using contrapositive or contradiction.
Almost always you should disprove a statement using a counter-example. (Sometimes
perhaps you could write down the negation and prove that.)
• If you’re trying to prove a statement, first try direct proof.
a. Prove or disprove: If S, T are wide, then S [ T is wide.
b. Prove or disprove: If S, T are narrow, then S \ T is narrow.
c. Prove or disprove: If a, b, c are integers and a | b and b | c, then a | c.
• If the statement is “If P , then Q” and you want to prove it, try assuming P and deducing
Q. Sometimes it is much easier to work with “not Q” than to work with P . In those cases,
you might try using contrapositive.
a. Prove or disprove: Every wide set is infinite. (First rewrite this as an if-then statement.
Try both a direct proof and a proof by contrapositive.)
• Another option for proving “If P , then Q” statements is to use contradiction. In this case,
you would assume P and not Q are both true, and attempt to arrive at a contradiction.
Very important: The negation of P ) Q is P ^ ¬Q.
a. Prove or disprove: If a, b are integers with b 2, then either b - a or b - a + 1.
• If the statement you’re trying to prove is about all positive integers (or something similar),
you might try to prove the statement using induction (or strong induction). It usually
doesn’t make sense to prove a statement about all real numbers, etc. using induction.
a. Prove or disprove: If k 2 is an integer and if a1 , . . . , ak are k odd integers, then their
product a1 · · · ak is also odd.
b. Prove or disprove: If x > 1 is a real number and k 1 is an integer, then (1 + x)k
1 + kx. (Does it make sense to try induction on k or on x?)
• To prove an “if and only if” statement, you have to prove two directions. To disprove an
“if and only if” statement, you only have to disprove one of the two directions.
• We saw a few famous proofs in class. You don’t need to memorize the proofs word-forword, but you should understand how they work:
a. There are infinitely many prime numbers.
p
b. The number 2 is irrational.
c. The set R is uncountable.
• Here are some challenge questions.
a. Prove orp disprove: If x, y > 0 are irrational, then xy is also irrational. (Hint. We have
p 2
p
that 2 is either rational or irrational. If it’s irrational, raise it to the 2 power again.)
b. Prove or disprove: If a set S is flipped, then its complement Z \ S is not flipped.
c. Prove or disprove: The function f (x) : R ! R given by x 7! x3 is one-to-one.