FORM TWO MATHEMATICS STUDY NOTES TOPIC 10-11.

TOPIC 10: SETS

Sets
Forget everything you know about number and forget that you even know what a number is. This is where mathematics starts. Instead of mathematics with numbers we will think about mathematics with things.
The word set means collection of related things or objects. Or, things grouped together with a certain property in common. For example, the items you wear: shoes, socks, hat, shirt, pants and so on. This is called a set. A set notation is simple, we just list each element or member (element and member are the same thing), separated by comma, and then put some curly brackets around the whole thing. See an example below:
The Curly brackets are sometimes called “set brackets” or “Braces”.
Sets are named by capital letters. For example; A = {1, 2, 3, 4 …} and not a = {1, 2, 3, 4, …}.
To show that a certain item belongs to a certain set we use the symbol .
For example if set A = {1, 2, 3, 4} and we want to show that 1 belongs to set A (is an element of set A) we write 1∈A.
To show the total number of elements that are in a given set, say set A, we use the symbol n(A). Using our example A = {1, 2, 3, 4}, then, the total number of elements of set A is 4. Symbolically , we write n(A) = 4
A Set
Define a set
We describe sets either by using words, by listing or by Formula. For example if set A is a set of even numbers, we can describe it as follows:
  1. By using words: A = {even numbers}
  2. By listing: A = {2, 4, 6, 8, 10,...}
  3. By Formula: A = {x: x = 2n, where n = 1,2,3,…} and is read as A is a set of all x such that x is an even number.
Example 1
Describe the following set by listing: N is a set of Natural numbers between 0 and 11
Solution
N = {1,2,3,4,5,6,7,8,9,10}
Example 2
Write the following named set using the formula: O is a set of Odd numbers:
Solution
O = {x: x = 2n – 1, whereby n = 1,2,3….}
Example 3
Write the following set in words: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Solution
W = {whole numbers} or W is a set of whole numbers.
The Members of a Set
List the members of a set
The objects in a set are called the members of the set or the elements of the set.
A set should satisfy the following:
  1. The members of the set should be distinct. (not be repeated)
  2. The members of the set should be well-defined. (well-explained)
Example 4
In question 1 to 3 list the elements of the named sets.
  1. A={x: x is an odd number <10}
  2. B={days of the week which begin with letter S}
  3. C={prime numbers less than 13}
Solution
1. A={1,3,5,7,9}
2. B={Saturday,Sunday}
3. C={2,3,5,7,11}
Naming a Set
Name a set
To describe a small set, we list its members between curly brackets {, }:
  • {2, 4, 6, 8}
  • { England, France, Iran, Singapore, New Zealand }
  • { David Beckham } {}
  • (the empty set, also written ∅)
We write a ∈ X to express that a is a member of the set X. For example 4 ∈ {2, 4, 6, 8}. a /∈ X means a is not a member of X.
Differentiate Sets by Listing and by Stating the Members
Distinguish sets by listing and by stating the members
By Stating the members: A = {even numbers}
By listing: A = {2, 4, 6, 8, 10,...}

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