Download Section 0.1 Sets of Real Numbers

Section 0.1 Sets of Real Numbers
Sets of Real Numbers
A set is a collection of objects. Usually written with {}.
Ex: The set of numbers one, two, and three: {1, 2, 3}. This is called roster notation. We
can also write sets in set-builder notation by giving a rule to tell which elements are in the set:
{x | x is 1, 2 or 3}
Important Sets of Numbers:
• Natural Numbers – The numbers we use for counting {1, 2, 3, 4, 5, 6,....}.
• Whole Numbers – The set of natural numbers plus the number 0: {0, 1, 2, 3, 4, 5, 6, ...}.
• Integers – The set of whole numbers and their negatives: {...,−6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, ...}
– Even Integers – Integers exactly divisible by 2: {..., −12, −10, −8, −6, −4, −2, 0, 2, 4, 6, 8, 10, 12, ...
– Odd Integers – Integers that are not exactly divisible by 2: {..., −13, −11, −9, −7, −5, −3, −1, 1, 3,
• Prime Numbers – A prime number is a natural number greater than 1 that is divisible only
by itself and 1:={2, 3, 5, 7, 11, 13, 17, 19, ...}
• Composite Numbers – A composite number is a natural number greater than 1 that is not
prime: {4, 6, 8, 9, 10, 12, 14, 15, 16, ...}
• Rational Numbers – Fractions that have an integer numerator and nonzero integer denominator: { ab |a is an integer and b is a nonzero integer}
• Irrational Numbers – Non-terminating, non-repeating decimals.
• Real Numbers – Any number that is rational or irrational: {x | x is a rational or irrational
number}
When every element of a set A is included in a set B, we say A is a subset of B.
Ex: The set of Integers is a subset of the set of Rational Numbers.
See Figure 0-1 in the text to see the relationships of the sets defined above.
a.
b.
c.
d.
e.
√
Ex: Given the set {−5, −4, − 32 , 0, 1, 2, 2, 2.75, 6, 7}.
List the natural numbers. 1, 2, 6, 7
List the whole numbers. 0, 1, 2, 6, 7
List the prime numbers. 2, 7
List the even integers. −4, 0, 2, 6
List the rational numbers. −5, −4, − 32 , 0, 1, 2, 2.75, 6, 7
Properties of Real Numbers
Real numbers have the following properties: (Assume a, b, and c are real numbers).
• Associativity for Addition and Multiplication:
(a + b) + c = a + (b + c)
(ab)c = a(bc)
• Commutativity for Addition and Multiplication:
a+b=b+a
ab = ba
• The Distributive Property of Multiplication over Division:
a(b + c) = ab + bc
a(b − c) = ab − ac
• The Double Negative Rule:
−(−a) = a
Graphing Sets of Real Numbers
We can graph subsets of the real numbers on a number line, with positive numbers to the right of
0 and negative numbers to the left of 0.
NOTE: The number 0 is neither positive nor negative.
The point associated with each number is called the graph of the number and the number is
called the coordinate of its point.
Each point on the number line corresponds to exactly one real number coordinate and each real
number corresponds to exactly one point on the number line.
Inequality Symbols
Symbol
6=
<
>
≤
≥
≈
Read as
Examples
is not equal to
5 6= 8
is less than
12 < 20
is greater than
15 > 9
is less than or equal to
25 ≤ 25
is greater than or equal to √19 ≥ 19
is approximately equal to
2 ≈ 1.414
NOTE: 15 < 23 is the same as 23 > 15.
Intervals
When graphing inequalities on a number line, use a parenthesis to show an endpoint is not included
and a square bracket to indicate an endpoint is included. For example, in the inequality x > 2, the
endpoint 2 is not included, while in the inequality x ≥ 5, the endpoint 5 is included.
The portion of the number line representing the inequality is called an interval.
Note that for x > 2, the interval extends infinitely far to the right. We can write this inequality in interval notation as (2, ∞). We write x ≥ 5 as [5, ∞).
If an interval extends forever in one direction, it is called an unbounded interval. The two intervals
above are unbounded.
When two inequalities are written as a single expression, this is known as a compound inequality.
A compound inequality can be written as two separate inequalities.
Ex: 3 < x < 16 can be written as x > 3 and x < 16.
We can write this as the intersection of two intervals, (3, ∞) ∩ (−∞, 16), or as the interval (3, 16).
A bounded interval with no endpoints is an open interval. (3, 16) is an open interval.
A bounded interval with one endpoint is a half-open interval. [−4, 6) is a half-open interval.
An interval containing two endpoints is a closed interval. [−3, 9] is a closed interval.
The expression x ≤ −6 or x > 7 represents the union of two intervals. This can be written
as (−∞, −6] ∪ (7, ∞).
Absolute Value
The absolute value of a real number x is the distance on a number line between 0 and the point
with a coordinate of x.
If x is a real number, then |x| = x when x ≥ 0 and |x| = −x when x < 0.
In general, | − x| = |x|
NOTE: Remember that x is not always positive and −x is not always negative!!
Ex: Write without using absolute value symbols:
a. |6| 6
b. | − 3| 3
c. −| − 21| −21
Ex: Write without using absolute value symbols:
a. |8 − π| 8 − π
b. |π − 5| −(π − 5) = 5 − π
c. |x + 1| and x ≤ −2 −(x + 1)
Distance on a Number Line
If a and b are the coordinates of two points on the number line, the distance between the points is
d = |b − a|.
Ex: Find the distance on a number line between
a. 4 and 6 2
b. −3 and 5 8
c. −1 and −8 7