Intuitively, a set is an unordered collection of distinct objects (but formally we leave set and element undefined). We write x∈ S to indicate that x is an element of S.

We discussed some of the axioms of set theory, namely the axiom of the empty set, the axiom of extentionality, the pairing axiom and the ``baby version'' of the axiom of union. The full set of axioms is here. (We will discuss only eight of the ten axioms.) We constructed the numbers 0 = { }, 1 = {0}, 2 = {0, 1}, …. We defined the successor function S(X) = X ∪ {X}, and used it to define successors of numbers. We defined addition recursively by
(IC) n+0 = n, and
(RR) n+S(k) = S(n+k).

We pointed out that there are possible problems with the definition of addition if either of the following happens:
(1) 0=S(k) for some k, or
(2) S(k)=S(l) for some k ≠ l.
We then proved that 0 does not equal S(X) for any set X. We will later prove that S(X)≠S(Y) whenever X≠Y.

We proved that S(n) = n+1. We proved that 2+2=4.

(IC) n*0 = 0, and
(RR) n*S(k) = n*k + k.

(IC) n⁰ = 1, and
(RR) n^S(k) = n^k*n.

A proof that 2*2=4.

Subsets. Axiom of comprehension/subset. Axiom of power set. Definition of inductive set. Axiom of infinity. Definition of the natural numbers as the set of elements common to every inductive set. Axiom of foundation. A proof that S(X)≠S(Y) whenever X≠Y.

We gave examples of direct proof.

Boolean algebra/Boolean logic.
We defined simple and compound propositions. We gave truth tables for ∧ (and), ∨ (or), ¬ (not), → (implies), ↔ (if and only if). We defined logical equivalence, tautology and contradiction.

We showed that every proposition is equivalent to a proposition expressed using only ∧, ∨ and ¬.

We showed the equivalence of (H→ C), ((¬ C)→(¬H)), and ((H∧(¬C))→F). We used these equivalences to compare direct proof with proof of the contrapositive and proof by contradiction. We proved the following theorem in each of the three ways:

Thm. If 0<x<1, then x²<x.

Read Section 10.
Quantifiers. We introduced ∀ and ∃. We examined some simple statements involving quantifiers. We defined predicate.

Quantifier games: we discussed how to determine whether a sentence written in prenex form is true in a structure.

We began discussing mathematical induction. The structure of a proof of a theorem using mathematical induction is:

Thm. A_n is true for all n∈ℕ.
Proof:
(Basis of induction, n=0) Prove A₀.
(Inductive step) Assume that A_n is true. Prove A_n+1. □

We used the method to prove:

Thm. 0+1+2+…+n=n(n+1)/2 is true for all n∈ℕ.
Proof:
(Basis of induction, n=0) We proved 0 = 0(0+1)/2.
(Inductive step) We assumed that
0+1+2+…+n=n(n+1)/2
holds, and proved that
0+1+2+…+n+(n+1)=(n+1)(n+2)/2
also holds. □

We continued discussing mathematical induction. We described the method of ordinary induction. We illustrated ordinary induction by proving:

Thm. n<2ⁿ for all n∈ℕ.

In our proof we noticed that the following lemma would be helpful:

Lemma. n+1≤2n for all n∈ℕ.

But then we noticed that the lemma is only true when n≥1. We modified the lemma statement to

Lemma. (Corrected statement.) n+1≤2n for all natural numbers n≥1.

and proved the lemma. Then we had to go back and modify the proof of the theorem, by proving the cases n=0 and n=1 in the basis step, in order to apply the lemma.

We described the method of strong induction. We illustrated strong induction by proving that every natural number greater than one is a product of primes.

In the second half of the lecture we discussed how induction is almost always needed to prove statements about recursively defined objects. To illustrate this, we proved some laws of addition by induction.

In preparation for our study of counting, we introduced enough terminology to be able to use the word "bijection". The new terminology includes: Cartesian product, relation (=subset of Cartesian product), function (= relation satisfying the Function Rule), injection (= one-to-one function), surjection (=onto function), bijection (= one-to-one and onto function). Function notation includes: f:A→B, f(a)=b, f:ℝ→ℝ:x↦x².

(i) f:A→B is a bijection.
(ii) Every element of A occurs as the first coordinate of a pair in f exactly once, and every element of B occurs as the second coordinate of a pair in f exactly once.
(iii) f^-1:B→A is a bijection.
(iv) f^-1:B→A is a function.

We defined |A|=|B|, |A|=n, finite, and infinite. We showed

(i) |A|=|A|.
(ii) If |A|=|B|, then |B|=|A|.
(iii) If |A|=|B| and |B|=|C|, then |A|=|C|.

and stated without proof that

(iv) If m, n∈ℕ and |m|=|n|, then m=n

We derived from the above that the cardinality of a finite set is uniquely determined.

We discussed the Sum Rule and the Product Rule.

Review Sheet for the Midterm Exam.

We recalled the proof that, if |A|=n and |B|=m, the number of functions f:A→B is mⁿ. We then proved that

(i) The number of injective functions from A to B is
(m)_n := m*(m-1)***(m-n+1).

(ii) The number of bijective functions from A to B is 0 if |A|≠|B| and (m)_m = m! if |A|=|B|=m.

(iii) The number of ways to linearly order an m-element set is m!.

We defined the binomial coefficient, C(n;k), to be the number of k-element subsets of an n-element set. We showed that

C(n;k) = (n)_k/k! = n!/(k!*(n-k)!).

C(n;k) = (n)_k/k! = n!/(k!*(n-k)!).

We derived the recursion for binomial coefficients (initial conditions + Pascal's Identity). We discussed Pascal's Triangle.

[NOTE: all the ``definitions'' written in parentheses in (i)-(iv) contain errors.]

We showed that a number of counting problems can be solved with the same formula:

1. How many terms are in the multinomial expansion of (x₁+…+x_n)^k?
2. For a given n and k, how many solutions in ℕ are there to
e₁+…+e_n = k?
3. How many ways are there to distribute k identical objects to n distinct recipients?
4. How many strings of k zeros and n-1 ones are there?
5. How many k-element multisets may be formed from an n-element source set?

Before posing problem 5, we defined a multiset (with finite multiplicities) to be a pair M = (S,m) where S is a source set and m:S→ℕ is a multiplicity function. The size of this multiset is defined to be &Sigma_{x∈ S} m(x). We introduced a notation for n-multichoose-k, which is the number that solves each of the above problems. We derived a formula for n-multichoose-k. We gave a recursion for the multichoose numbers. We solved some practice problems.

After the quiz, we discussed extending the Sum Rule to a rule for computing the union of two nondisjoint sets. The formulas we obtained were:

|A∪B| = |A-B|+|A∩B|+|B-A|, and
|A∪B| = |A|+|B|-|A∩B|.

By induction we can extend the second formula to a formula, called the principle of inclusion and exclusion, to compute the size of a union of n arbitrary sets. Details are in this handout. Solutions are in this document.

We reviewed and extended terminology about functions:

Old definitions:
1. f:A→B is a function from A to B.
2. A = dom(f) is the domain of f.
3. B = codom(f) ( = ran(f)) is the codomain of f (or the range of f).
4. f ^-1(b) = {a∈A | f(a)=b} is the fiber of f over b (or the inverse image of b).
5. im(f) = {b∈B | ∃a∈ A (f(a)=b)} is the image of f. (This is the set of elements of B whose fiber is nonempty.)

New definitions:
6. coim(f), the coimage of f, is the set of nonempty fibers of f.
7. If I is a subset of B, then the inclusion function is the function inc:I→B:x↦x.
8. If P is a partition of A, then the natural function is the function nat:A→P:x↦[x]. ([x] is the cell of P that contains x.)

We proved some theorems.

Thm.
1. If f:A→B is a function, then the image of f is a subset of B.
2. Conversely, if I is a subset of B, then I is the image of some function, namely the inclusion function inc:I→B.

Thm.
1. If f:A→B is a function, then the coimage of f is a partition of A.
2. Conversely, if P is a partition of A, then P is the coimage of some function, namely the natural function nat:A→P.

Thm. (Canonical Factorization of a Function)
If f:A→B is a function, then there is a unique function f:coim(f)→im(f) such that f = inc∘f∘nat. The function f is defined by f([x]) := f(x), and it is a bijection from coim(f) to im(f). (f is called the induced function.)

We discussed the bijection between equivalence relations ( = abstractions of kernels of functions) and partitions ( = abstractions of coimages of functions). We began a construction of the integers.

We discussed the idea of a graph. We explained the solutions to the Seven Bridges of Königsberg Problem and the Icosian Problem (= Around the World game).

We began discussing graph theory terminology. We proved that the sum of the degrees of the vertices of a finite graph is twice the number of edges, and derived from this that any finite graph has an even number of edges of odd degree. We observed that any pair of distinct points connected by a walk or trail is also connected by a path.

The relation "connected to" is an equivalence relation on the vertex set of a graph. The equivalence classes are the connected components.

Euler's Thm. A connected (multi)graph has a closed Eulerian trail iff all vertices have even degree and has an open Eulerian trail iff exactly two vertices have odd degree.

Ore's Thm. A graph G=(V,E) with |V|≥3 that satisfies

(*) whenever u, v∈V are nonadjacent, deg(u)+deg(v)≥n

has a Hamiltonian cycle.

Quiz 8.

We completed the proof of

Ore's Thm. A graph G=(V,E) with |V|≥3 that satisfies

(*) whenever u, v∈V are nonadjacent, deg(u)+deg(v)≥n

has a Hamiltonian cycle.

We began to discuss isomorphism of graphs.