1. Mathematical Notions and Terminology
Lecture 2
Section 0.2
Fri, Sep 1, 2006

2. Functions and Relations
A function associates every element of its domain with exactly one element of its range, or codomain.

3. Functions and Relations
Let f : A  B be a function.
f is one-to-one if
f(x) = f(y)  x = y.
Equivalently,
x  y  f(x)  f(y).

4. Functions and Relations
F is onto if for every y  B, there is x  A such that f(x) = y.

5. A One-to-one Correspondence
A hallway has 100 lockers, numbered 1 through 100 .
All 100 lockers are closed.
There are 100 students, numbered 1 through 100 .

6. A One-to-one Correspondence
For each k, student k’s instructions are to reverse the state of every k-th locker door, starting with locker k.
If all 100 students do this, which lockers will be left open?

7. A One-to-one Correspondence
Which students should be sent down the hall so that exactly the prime-numbered lockers are left open?
Is it possible to leave any specified set of lockers open?

8. A One-to-one Correspondence
Let L be the set of all lockers and let S be the set of all students.
Let f :(S)  (L) be defined as
f(A) is the set of locker doors left open after the students in A have gone down the hall.

9. A One-to-one Correspondence
Show that f is a one-to-one correspondence.

10. Functions and Relations
A subset of A  A is called a (binary) relation on A.

11. Functions and Relations
A binary relation R is an equivalence relation if it is
Reflexive: (x, x)  R.
Symmetric: (x, y)  R  (y, x)  R.
Transitive: (x, y)  R and (y, z)  R  (x, z)  R.
for all x, y, z  R.

12. Graphs
A graph consists of a finite set of vertices and a finite set of edges.
In a directed graph, each edge has a direction from one vertex to another.

13. Graphs and Relations A graph may be used to represent a relation.
Draw a vertex for every element in A.
If a has the relation to b, then draw an edge from a to b.

14. Strings and Languages
An alphabet is a finite set of symbols, typically letters, digits, and punctuation.
A string is a finite sequence of symbols from the alphabet.
The empty string  is the unique string of length 0.

15. Strings and Languages Let the alphabet be  = {a, b}. Some strings: 
aaa
abb
abbababbabbbababab

16. Lexicographical Order
Assume that the symbols themselves are ordered.
Group the strings according to their length.
Then, within each group, order the strings “alphabetically” according to the ordering of the symbols.

17. Lexicographical Order
Let the alphabet be  = {a, b}.
The set of all strings in lexicographical order is
{, a, b, aa, ab, ba, bb, aaa, …, bbb, aaaa, …, bbbb, …}