Discrete Mathematics (Math 120.004)
Lecture 1
3 Denition
An Object X is called the term being dened provided it satises specic conditions.
Examples:
1. Use the concept of integers Z, we can dene the t
Discrete Mathematics (Math 120.004)
Lecture 2
6 Counterexample
To disprove a statement If A, then B, we only need to nd an example where A is true but B is false.
Examples:
1. Disprove: If a, b and c
Discrete Mathematics (Math 120.004)
Lecture 3
8 Lists
Denition
A list is an ordered sequence of objects. The number of elements in a list is called its length. A list of
length 2 is also called an ord
Discrete Mathematics (Math 120.004)
Lecture 4
10 Sets I: Introduction, Subsets
1. Denition
A set is a repetition-free, unordered collection of objects. An object that belongs to a set is called an
ele
Discrete Mathematics (Math 120.004)
Lecture 5
11 Quantiers (Continued)
4. Negating Quantied Statements
(1) =
(2) =
(3) (A and B) = (A) or (B)
(4) (A or B) = (A) and (B)
Examples:
Disprove the follow
Discrete Mathematics (Math 120.004)
Lecture 6
14 Relations
A relation is a set of ordered pairs.
Notation: x R y (x, y) R. x R y (x, y) R.
/
/
We say R is a relation on a set A provided R A A, and R i
Discrete Mathematics (Math 120.004)
Lecture 7
16 Partitions
A partition of a set A is a set of nonempty, pairwise disjoint sets whose union is A.
(Counting equivalence classes) Let R be an equivalence
Discrete Mathematics (Math 120.004)
Lecture 8
17 Binomial Coecients (continued)
(Pascals Identity) Let n and k be integers with 0 < k < n. Then
n
k
=
n1
n1
+
.
k1
k
(1)
Examples:
1. What is the coecie