DISCRETE MATH: LECTURE 1
DR. DANIEL FREEMAN
1. In the Beginning.aka Chapter 1.1
1.1. The Trinity.
A Universal Statement says that a certain property is true for all elements in a
set. (for all )
A Conditional Statement is an if-then statement, that is,
DISCRETE MATH: LECTURE 13
DR. DANIEL FREEMAN
1. Chapter 5.4 Strong Mathematical Induction
Proving a statement by either mathematical induction or strong mathematical induction
is a two step process the rst step is called the basis step, the second step is
DISCRETE MATH: LECTURE 15
DR. DANIEL FREEMAN
1. Chapter 6.1 Set Theory: Definitions and the Element Method of Proof
continued!
Denition (ordered n-tuple). Let n be a positive integer and let x1 , x2 , ., xn be n elements. (x1 , x2 ) is called an ordered p
DISCRETE MATH: LECTURE 12
DR. DANIEL FREEMAN
1. Chapter 5.2 and 5.3 Mathematical Induction I and II
Proving a statement by mathematical induction is a two step process the rst step is
called the basis step, the second step is called the inductive step.
Pr
DISCRETE MATH: LECTURE 15
DR. DANIEL FREEMAN
1. Chapter 6.1 Set Theory: Definitions and the Element Method of Proof
Recall that a set is a collection of elements.
Some examples of sets of numbers are:
Z = cfw_., 2, 1, 0, 1, 2, . is the set of integers.
DISCRETE MATH: LECTURE 10
DR. DANIEL FREEMAN
1. Chapter 4.6 Indirect Argument: Contradiction and Contraposition.
(continued from Lecture 9)
Recall:
A statement and its contrapositive are given below:
x D, if P (x) then Q(x).
x D, if Q(x) then P (x).
A s
DISCRETE MATH: LECTURE 8
DR. DANIEL FREEMAN
1. Chapter 4.2 Direct Proof and Counterexample 2: Rational Numbers
Denition. A real number r is rational if and only if it is equal to the quotient of two
integers with a nonzero denominator. A real number is ir
DISCRETE MATH: LECTURE 3
DR. DANIEL FREEMAN
1. Chapter 2.2 Conditional Statements
If p and q are statement variables, the conditional of q by p is If p then q or p
implies q and is denoted p q . It is false when p is true and q is false; otherwise
it is
DISCRETE MATH: LECTURE 6
DR. DANIEL FREEMAN
1. Chapter 1 review
1) a.
b.
c.
d.
e.
Does 3 = cfw_3?
Is 3 cfw_3?
Is 3 cfw_3?
Does cfw_3 = cfw_3, 3, 3, 3?
Is cfw_x Z|x > 0 cfw_x R|x > 0?
2) a. When does (a, b) = (c, d)?
b. If A = cfw_1, 2 and B = cfw_x, y, c
DISCRETE MATH: LECTURE 2
DR. DANIEL FREEMAN
1. Chapter 2.1 Logical Form and Logical Equivalence
1.1. Deductive Logic.
An Argument is a sequence of statements aimed at demonstrating the truth of
an assertion.
The assertion at the end of the sequence is c
DISCRETE MATH: LECTURE 5
DR. DANIEL FREEMAN
1. Chapter 3.3 Statements with Multiple Quantifiers
If you want to establish the truth of a statement of the form
x D, y E such that P (x, y )
your challenge is to allow someone else to pick whatever element x
DISCRETE MATH: LECTURE 11
DR. DANIEL FREEMAN
1. Chapter 5.2 Mathematical Induction I
Principle of Mathematical Induction
Let P (n) be a property that is dened for all integers n. Suppose the following two
statements are true.
1. P (a) is true.
2. For all