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 terms divisible, odd, even, prime, and composite.
2. Use
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 are positive integers such that a|(bc), then a|b or a|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 ordered pair. A list of length 0 is called the empty list.
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
element of the set. The notation for membership in a set i
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 following statement: x Z, y Z, such that x2 y 2 = x + y and x
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 is a relation from set A to set B provided
R A B.
Invers
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 relation on a nite set A. If all the equivalence class
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 coecient of x3 in (2x + 1)6 ?
2. Use the Binomial Theorem to