1A - Organising numbers

Learning intentions:

In this section we will examine:

    1. The real number system.
    2. Set notation.
    3. Interval notation.

The real number system

The real number system (ℝ) is the collection of numbers you will be most familiar with. The real number system contains the set of all rational and irrational number. Please note:

    • ℝ+ contains all positive numbers in the real number system (not zero, 0).
    • ℝ- contains all negative numbers in the real number system (not zero, 0).

A rational number (ℚ) is any number that can be expressed as a fraction. Rational numbers include the following sets:

    • Natural numbers, or counting numbers, ℕ = {1,2,3,4, ...}
    • Whole numbers, 𝕎 = {0,1,2,3,4, ...}
    • Integers, ℤ = {... ,-4,-3,-2,-1,0,1,2,3,4, ...}
    • And any other number that can be expressed as a fraction of two numbers (a and b):

We use bold-faced capital letters to denote sets of numbers. For example, the real number set can be described with ℝ.

In other mathematics subjects you may have come seen the complex number system (ℂ). This set of numbers includes the imaginary number i:

An irrational number (ℚ') is any number that cannot be expressed as a fraction. Irrational numbers include, but are not limited to, the following numbers:

    • π - pi.
    • e - the natural number.
    • - the golden ratio.
    • Surds, including √2 or √5.

A Venn-diagram can be used to visually display the sets involved in the real number system:

\varphi

The apostrophe (ℚ') is used to denote the complement of a set; that is, all numbers not contained within ℚ.

The intersection of two sets contains all elements that are common to both sets. The intersection of A and B is: A ∩ B

The Real Number System

Figure 1 - The Real Number System (ℝ)

Set and interval notation

A set is simply a collection of objects. The objects within a given set are known as elements.

  • If x is an element of set A then we can represent this mathematical as x ∈ A
  • If x is not an element of set A then we can represent this mathematical as x ∉ A.

A set B is a subset of A if and only if x ∈ B implies x ∈ A.

    • To indicate that B is a subset of A we can represent this mathematically as B ⊆ A.

Intersection

The set of elements contained in both sets A and B is called the intersection.

    • We denote the intersection between A and B mathematically as: A ∩ B
    • Note: A ∩ B is equivalent to B ∩ A.
Intersection

Figure 2 - The intersection of A and B (A ∩ B).

Null sets

If sets A and B have no elements in common; that is, there are no elements common to both sets then the intersection is an empty or null set (∅). Furthermore, on the Venn diagram would have no overlap between the two circles representing A and B (see below).

  • Mathematically we can denote the disjoint (null set) A and B as A ∩ B = ∅.

Mutually exclusive sets/events

If two sets (or events in probability), A and B have no elements in common then they are considered mutually exclusive.

Figure 3 - Two mutually exclusive events A and B (A ∩ B = ∅).

Union

The set of elements contained in A and B; that is, all elements in A and all the elements in B, is called the union. In the case of a union, the elements do not have to be in both A and B

(just in one of them).

  • Mathematically we express the union of sets A and B as: A ∪ B
Union

Figure 4 - The union of A and B (A ∪ B).

A union can be used to include additional elements within a set. For example, consider the set ℝ+ which includes all positive numbers in the real number system; however, for some reason you wish to include zero - we can write this as ℝ+ ∪ {0}.

When working with sets of numbers, we can exclude certain elements to describe a new set of numbers. To do this we use a backslash (\) and list the excluded elements in a pair of {curly brackets}. For example, consider the real number set contains which contains the number zero; however, for some reason you may wish to exclude zero - we can write this as: ℝ\{0}.

Complementary sets

The complementary set of A is all of the element in not in set A.

    • Mathematically we represent the complement of A as A'.
    • The union of two complementary sets, A ∪ A', will contain all elements in the system.

Figure 5 - The complement of A (A').

Creating your own sets

In mathematics you are often required to describe a set of number that is not already defined by a convenient symbol (such as the ones previously discussed). In this case you often need to describe what numbers your set falls between and indicate whether or not the endpoints are include or excluded.

Set notation

    • The statement below states that x is any number greater than a and less than b.

a < x < b

    • The statement below state that x is any number greater than or equal to a and less than or equal to b.

a ≤ x ≤ b

Interval notation

    • The statement below states that x is any number greater than a and less than b. Here, (rounded brackets) are used to signify the number is not included.

x ∈ (a,b)

    • The statement below state that x is any number greater than or equal to a and less than or equal to b. Here, [square brackets] are used to signify the number is included.

x ∈ [a,b]

1A - Exercises:

    1. ...

Success criteria:

You will be successful if you can:

    1. Recall and express common sets of numbers including the real number system and integers.
    2. Identify rational and irrational numbers.
    3. Understand the mathematical meaning of an intersection, union and complementary set.
    4. Express a collection of numbers using set and/or interval notation.