Set of Real Numbers
A set is a well defined collection of objects, items or elements. In Mathematics, we usually label a set with a capital letter, eg. A, B, C etc, and the elements are wrapped inside of a curly bracket.
Below is an example of a set A whose elements are odd numbers less that 10:
\[A = \{1, 3, 5, 7, 9\}\]
You say, "The set A equals 1, 3, 5, 7, 9."
Every set must be well defined in such a way that we can describe the members inside the given set. For instance, a set of oranges should have only oranges and no other fruits.
Examples of sets are: A set of chairs, a set of students in basic 9, a set of footballs, etc.
There are 3 basic ways that we can describe a set. They are:
1. Word description:
Where we use sentences and statements or phrases to describe given sets.
Eg. 1. \(B =\) \(\{\)whole numbers from 20 to 30\(\}\)
\(\hspace{0.8cm}\) 2. \(M =\) \(\{\)factors of 63\(\}\)
\(\hspace{0.8cm}\) 3. \(K =\) \(\{\)odd numbers less than 13\(\}\)
2. Listing:
With listing, the elements are listed with a comma (,) separating each of the elements in the
set.
Let's list the elements in the sets described above:
Eg. 1. \(B =\) \(\{\)20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30\(\}\)
\(\hspace{0.8cm}\) 2. \(M =\) \(\{\)1, 3, 7, 9, 21, 63\(\}\)
\(\hspace{0.8cm}\) 3. \(K =\) \(\{\)1, 3, 5, 7, 9, 11\(\}\)
3. Set builder notation.
Set builder notation is when we use mathemtical and logical notations to build sets.
Let's use set builder notation to build the sets listed above:
Eg. 1. \(B =\) \(\{x: 20 \le x \le 30, x\) is a whole number\(\}\)
\(\hspace{0.8cm}\) 2. \(M =\) \(\{x:x\) \(\in\) factors of 63\(\}\)
\(\hspace{0.8cm}\) 3. \(K =\) \(\{x:x < 13, x \in\) odd numbers\(\}\)
Note
\(\mathbf{x:x < 13}\): You say "x such that x is less than 13."
The numbers, objects, items, inside of the curly bracket are called elements, or members. The mathematical symbol for element is \(\in\).
Given that the set \(A = \{1, 3, 5, 7, 9\}\), you can therefore say that:
\(\hspace{0.7cm}\Rightarrow 1 \in A\)
\(\hspace{0.7cm}\Rightarrow 3 \in A\)
\(\hspace{0.7cm}\Rightarrow 5 \in A\)
\(\hspace{0.7cm}\Rightarrow 7 \in A\)
\(\hspace{0.7cm}\Rightarrow 9 \in A\)
The number of elements within any given set, say set A, is given as below:
\[\mathbf{n(A) = n}\]
Hence, for our set A above, \(n(A) = 5\), that is, the number of elements within the set A is 5.
Example 1
What is the number of elements in the set \(B=\) \(\{\)odd numbers less than 10\(\}\)?
Solution
Odd numbers \(\Rightarrow \{1, 3, 5, 7, 9, 11, 13, 15,...\}\)
Odd numbers less than 10 \(\Rightarrow \{1, 3, 5, 7, 9\}\)
\(\Rightarrow B =\{1, 3, 5, 7, 9\}\)
\(\Rightarrow n(B) = 5\)
\(\therefore\) the number of elements in the set B is 5.
A subset can simply be defined as a set that belongs to a bigger set. Subsets are formed by taking or listing a part of the elements inside of a parent-set in another curly bracket.
A subset can contain as many elements as there are in the bigger set (or parent-set) under consideration.
Every element or combination (group) of elements within any given set can be described as a subset of the set.
For example, given the set \(A = \{1, 3, 5, 7, 9\}\), some of the subsets or smaller sets that can be formed from this set are \(\{1, 3\}\), \(\{1, 3, 5\}\), \(\{5, 7, 9\}\), \(\{9\}\), etc.
Think and come up with other subsets that can be formed from this set A and show it to your friends.
Example 1
Kofi listed the prime numbers between 1 and 20 in a set A. Then he listed the prime numbers between 1 and 10 in another set B. Describe the set B in terms of the set A.
Solution
Set \(A = \{\)prime numbers between 1 and 20\(\}\)
\(\Rightarrow A = \{2, 3, 5, 7, 11, 13, 17, 19\}\)
Set \(B = \{\)prime numbers between 1 and 10\(\}\)
\(\Rightarrow B = \{2, 3, 5, 7\}\)
Since all the elements in \(B\) can be found in \(A\), we can therefore say that, "Set B is a subset of set A."
And mathematically,
\(\Rightarrow \mathbf{B \subset A}\)
Number of Subsets \((2^n)\)
Given any set, say A, the total number of subsets \(n(A)\) that can be formed using the elements within the set can be calculated using the formular below:
\[\mathbf{Number\ of \ subsets = 2^n}\]
Where \(\mathbf{n}\) represents the number of elements in the given set.
A set of all the subsets that can be formed within any set is termed the power set.
Note:
In our number system, the biggest set of numbers that we will consider at our level is called the set of real numbers. The set of real numbers has other subsets.
We will now look at these sets and the relationships between them in the rest of this lesson.
1. The Set of Real Numbers \((\mathbf{\mathbb{R}})\) :
The first set we will learn about is the set of real numbers. This is the biggest set of all
the set we will consider.
The set of real numbers is the set that contains all rational \((Q)\) and irrational numbers \((I)\). Since it is the largest set, all the subsets of these two mentioned sets in the definition (rational and irrational) are also subsets of the set of real numbers.
2. Rational Numbers \((Q)\) :
Rational numbers are numbers that can be expressed as fractions or in the form \(\frac{a}{b}\).
They can also be thought of as fractions or decimals that terminate or repeat.
Eg. 0.5, \(\frac{1}{8}\), 68%, 3.75, 79, etc.
3. Irrational Numbers \((I)\) :
Irrational numbers are numbers that have non-repeating or terminating decimals. Examples of irrational
numbers are given as below:
\(\pi\), \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), etc.
Research:
So far in our mathematics lessons, we have been using \(\pi = \frac{22}{7}\). Which means we have been expressing
\(\pi\) as a rational number.
Research on why \(\pi\) is an element of the set of irrational numbers.
4. Intergers \((Z)\) :
Integers are defined as the set of negative and positive whole numbers. Remember, that this
set also includes the number zero (0).
E.g. \(Z=\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\).
5. Whole Numbers \((W)\) :
Whole numbers are the set of non-negative integers that include the number zero (0). The elements
within this set begin from zero and continues to infinity.
E.g. \(W=\{0, 1, 2, 3, 4, \dots\}\).
6. Natural Numbers \((N)\) :
Natural numbers are positive integer values. They do not include the number zero (0). Hence,
natural numbers are the numbers that we normally use to count, they are sometimes
termed as counting numbers.
E.g. \(N=\{1, 2, 3, 4, \dots\}\).
Mathematically, the above definitions imply:
\[N \subset W \subset Z \subset Q \subset \mathbb{R}\]
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