MATH 2200 Final Review
Thomas Goller
December 7, 2012
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Exam Format
The final exam will consist of 8-10 proofs. It will take place on Tuesday, December 11, from 10:30 AM 12:30 PM, in the usual room.
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Topics
Here is a (not necessarily complete) list of key topics.
(1) Types of Proofs
• Direct proof.
• Proof by contrapositive.
• Proof by contradiction.
• Proof by induction.
(2) Sets and functions.
• Key set definitions: set, subset, union, intersection.
• Key function definitions: function, injective, surjective, bijective, composition of functions, invertible.
• Key theorems: a function is invertible if and only if it is bijective.
(3) Number theory
• Key divisibility definitions: divide, prime factorization.
• Key modular arithmetic definitions: congruent, Fermat’s little theorem.
• Key theorems: division algorithm, fundamental theorem of arithmetic (prime factorization), linear
combination corollary to Euclidean algorithm.
(4) Graph theory
• Key definitions: graph, subgraph, union of subgraphs, intersection of subgraphs, walk, connected,
complete, simple, bipartite, planar.
• Key theorems: inductive construction of connected graphs, Euler’s formula.
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3
How to Prove a Theorem
(1) Think about whether the theorem is plausible. Check cases. Try to construct a counterexample. If the
theorem seems true, proceed to the next step.
(2) Write the assumptions (premises) at the top of your page and the conclusion you want to deduce at the
bottom.
(3) Try to use the assumptions to deduce the conclusion. This may involve:
• Direct proof. Apply definitions to the assumptions and maybe invoke some theorems to directly
deduce the conclusion. If this doesn’t work, consider:
• Proof by contrapositive. If the theorem is of the form “if P , then Q”, try to prove “if ¬Q, then
¬P ” instead.
• Proof by contradiction. Add the negation of the conclusion to your assumptions and try to derive
a contradiction.
• Proof by induction. If some integer associated to the theorem ranges over all Z≥0 (or Z≥1 , or Z≥2 ,
etc.), and if the theorem is easier to prove when that integer is small, then consider using induction
on that value.
(4) If you found an argument that yields the conclusion, write it out in precise crisp English sentences! Label
it “Proof” and draw your favorite symbol at the end.
(5) If nothing is working, think hard about whether the theorem is false! Try to construct a counterexample
based on where you are getting stuck in your attempted proofs. If you find a counterexample, write
“The theorem is false!” and give your counterexample.
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Example Problems
Beware! Many of the theorems are false! Before plunging into a proof, think about whether it is plausible.
Checking some cases is always a good idea. Many of the theorems come from lecture, homework, and quizzes.
Some of them (especially in the graph theory section) are quite difficult! Don’t feel pressured to do them
all, but familiarize yourself with the terms that keep popping up.
4.1
Induction
Theorem 1. Let n ∈ Z≥1 . Then:
(a) The sum of the first n positive integers is
n(n+1)
,
2
namely 1 + 2 + · · · + n =
n(n+1)
.
2
(b) The sum of the first n even positive integers is n(n + 1), namely 2 + 4 + · · · + 2n = n(n + 1).
(c) The sum of the first n odd positive integers is n2 , namely 1 + 3 + 5 + · · · + (2n − 1) = n2 .
(d) 1 · 1! + 2 · 2! + · · · + n · n! = (n + 1)! − 1.
(e) 12 − 22 + 32 − · · · + (−1)n−1 n2 = (−1)n−1 n(n+1)
.
2
(f ) 1 · 2 + 2 · 3 + · · · + n(n + 1) =
(g) 13 + 23 + · · · + n3 =
n(n+1)(n+2)
.
3
n2 (n+1)2
.
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2
4.2
Sets and functions
Theorem 2. Let n ∈ Z>0 . Then 2n =
(Hint: use the binomial theorem)
!n
j=0
" n#
j
.
Theorem 3. Let n ∈ Z≥0 . The number of subsets of a set with n elements is 2n .
Theorem 4. Let A
f
! B be a function.
(a) There is a subset A# ⊆ A such that restricting the domain of f to A# yields an injective function
A#
f
!B.
(b) There is a subset B # ⊆ B such that restricting the codomain of f to B # yields a surjective function
A
f
! B# .
Theorem 5. Let A, B, C be sets, let A
composition of f and g. Then:
f
! B and B
g
! C be functions, and let A
g◦f
! C be the
(a) If f and g are injective, then g ◦ f is injective.
(b) If f and g are surjective, then g ◦ f is surjective.
(c) If g ◦ f is injective, then f is injective.
(d) If g ◦ f is injective, then g is injective.
(e) If g ◦ f is surjective, then f is surjective.
(f ) If g ◦ f is surjective, then g is surjective.
4.3
Number theory
Theorem 6. Let n ∈ Z. Then n is even if and only if 2 | n if and only if n ≡ 0 (mod 2).
Theorem 7. Let m, n ∈ Z, let d ∈ Z≥1 , and let p ∈ Z≥2 be prime. Then:
(a) If n is odd, then mn is odd.
(b) If m and n are odd, then mn is odd.
(c) m and n are odd if and only if mn is odd.
(d) n is odd if and only if n2 is odd.
(e) n is odd if and only if nd is odd.
(f ) If d | n, then d | mn.
(g) If d | mn, then d | m or d | n.
(h) If p | mn, then p | m or p | n.
Theorem 8. There are infinitely many primes.
Theorem 9 (Key lemma for the Euclidean algorithm). Let a = bq + r, where a, b, q, r ∈ Z. Then gcd(a, b) =
gcd(b, r).
Theorem 10. Let a, b, c ∈ Z with a &= 0. Then
(a) If a | b and a | c, then a | (b + c).
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(b) If a | b, then a | bc for all integers c.
(c) If a | b and b | c, then a | c.
Theorem 11 (Theorem that makes computation modulo m easy). Let m ∈ Z>0 . If a ≡ b (mod m) and c
is any integer, then
a + c ≡ b + c (mod m)
and
ac ≡ bc (mod m).
Theorem 12. Let a ∈ Z≥0 and let D be the sum of the digits of a. Then 3 | a if and only if 3 | D.
4.4
Graph theory
Theorem 13. Let H be a subgraph of a graph G.
(a) If G is simple, then H is simple.
(b) If G is a complete graph, then H is a complete graph.
(c) If G is connected, then H is connected.
(d) If G is bipartite, then H is bipartite.
(e) If G is planar, then H is planar.
(f ) The number of connected components of H is less than or equal to the number of connected components
of G.
(g) The number of connected components of H is greater than or equal to the number of connected components
of G.
Theorem 14. Let H1 , H2 be subgraphs of a connected graph G. Then:
(a) H1 ∪ H2 is connected.
(b) If H1 ∩ H2 is non-empty, then H1 ∪ H2 is connected.
(c) H1 ∩ H2 is connected.
(d) If H1 ∪ H2 is non-empty, then H1 ∩ H2 is connected.
Theorem 15. Let G = (V, E, φ) be a graph.
!
(a)
v∈V deg(v) = 2 · |E|.
(b) The number of vertices in G with odd degree must be even.
" #
(c) |E| ≤ |V2 | .
" #
(d) If G is simple, then |E| ≤ |V2 | .
(e) If G is bipartite, then every walk that starts and ends at the same vertex has an even number of edges.
(f ) If G is bipartite, then |E| ≤
|V |2
4 .
(g) If G is simple and bipartite, then |E| ≤
|V |2
4 .
Here are some challenge problems (you will need to use Euler’s formula):
Theorem 16. Let G = (V, E, φ) be a planar graph. Set v = |V | and e = |E|. Let r be the number of regions
of a planar drawing of G. Then:
(a) The number of connected components of G is v − e + r − 1.
(b) If G is connected, simple, and v ≥ 3, then e ≤ 3v − 6.
(c) If G is connected, simple, v ≥ 3, and G has no subgraph isomorphic to the complete graph K3 , then
e ≤ 2v − 4.
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Hints to Example Problems
• Theorem 1:
• Theorem 2: proved in notes.
• Theorem 3: follows from Theorem 2.
• Theorem 4:
• Theorem 5: two of these are false!
• Theorem 6: just understand the definitions.
• Theorem 7: two of these are false! Some of these are very easy if you use congruences.
• Theorem 8-12: proved in notes.
• Theorem 13: four of these are false!
• Theorem 14: three of these are false!
• Theorem 15: two of these are false!
• Theorem 16 (a): use the fact that all connected components share the same “infinite” region.
• Theorem 16 (b): since G has no loops or multiple edges, every region has at least 3 edges on its
boundary. Every edge appears on the boundary of two regions.
• Theorem 16 (c): Like (b) except that every region has at least 4 edges on its boundary.
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