Fundamentals
Lesson 1.1
Terminology for The Real Numbers
Before we start this lesson, here is a short clip to give you a brief introduction of set notation.
http://www.onlinemathlearning.com/set-notation.html
The Real Number System
A set, informally, represents a collection of terms that share similar idea.
Then, the set of Real Numbers ( ) contains some important subsets.
The Counting Numbers (i.e. 1, 2, 3, etc…) consists of all Positive Integers ( ). In IB, adding
zero (0) to the set of Positive Integers forms the set of
(i.e. 0, 1, 2, 3, etc…). Using set
notation, we have
x |1, 2,3,... or
x | x 1 , where means element.
Unfortunately, there is no universal agreement on the definition of whole number and
natural number. Thus we will not be using any of these two words in IB.
By combining the Negative Integers ( ) with the set of , this new subset is called the
Integers,
(i.e. … -3, -2, -1, 0, 1, 2, 3, … ). However, there are more numbers than just
integers ( ) on the real number line. Any number that can be expressed as a ratio of integers
is called a rational number ( ). Note that the set of rational numbers also includes all
numbers in the set of integers ( ).
3
5
, 2 , 8, 1.62, 0.5, 0.3, 0.987654321, 0.121212...
7
9
A real number that cannot be expressed as a ratio of integers is called an Irrational Number.
Examples of rational number include:
Examples of irrational number (
) include: 0.121231234...,
5, , 3.1415926..., e
Eg1. Categorize the following into the indicated sets using the set notation.
a) {cow, duck, apple, banana, bat}
b) {1, 2, 3, 5, 7, 11}
Animal =
Even =
Fruit =
Prime =
Composite =
Subsets
A subset is a (small) set of elements that belong to another (bigger) set with more elements.
Using the example from the Real Number System, the set of Positive Integers is a subset of
the Rational Numbers. In IB notation, we have
. Further, a proper subset is a subset
that contains fewer elements than the other subset, thus, it is true that
since
contains fewer elements (no negative integers) than the
Eg2. Express the following statements using proper notation.
a) Given: X = All students in this class
b) Given:
Y = All students at WVSS
, we have
. In addition,
.
= All real numbers
= All positive integers
c) Given: P = Students enrolling in Math11 SL this year
C = Students who are either in Ms. Dobson’s D1B4 class or D2B1 class
Intersection & Union
In Eg1b, with P = {2, 3, 5, 7, 11} and E = {2}, then 2 P and 2 E . The element 2 is the
intersection, , between set P and set E. We would express it with P E 2 .
However, sometimes we want to indicate all elements that are in either or both sets. We
would express it with union, .
Eg3. Determine the following.
a)
b)
c)
d)
Interval Notation
In IB, intervals on the real number line can be bounded (closed) or unbounded (open or halfopen). Also, they are expressed differently. See the following examples.
Eg4. Express each inequality using interval notation and on a number line.
a) 3 x 5
Practices:
p.6
b) x 4
# 1 – 42
c) | x | 2
d) | x | 2
e) x
0
