Download IGCSE Mathematics – Sets and set notation

IGCSE Mathematics – Sets and set notation
SETS AND SET NOTATION
We often deal with groups or collection of objects, such a set of books, a group of students, a list of cities in a
country, a collection of baseball cards, etc.
Sets may be thought of as a mathematical way to represent collections or groups of objects.
Although it was invented at the end of the 19th century, set theory is now an essential foundation for various
other topics in mathematics.
Definition: A set is a well defined group of objects or symbols.
(or: A set is a collection of distinct objects, considered as an object in its own right. )
Sets are named with capital letters: A, B, C, S, …
Elements
The objects in sets are called elements (or members).
If an element e belongs to set S this is represented as eS.
If an element f does not belong to set S this is represented as fS.
Writing sets
1. Listing method
All of the elements of a set are written.
A = {1, 3, 5, 7, 9}
2. Set - builder method
A typical element is named, along with its description, such as:
B = {x | x is an odd number from 1 to 10}
The vertical bar | is read: ‘‘such that’’, and it also can be written as :
3. Venn diagram
A Venn diagram is used for graphical representation of the set as a collection of its objects.
Example 1
This is a typical set, consisted of numbers 1, 2 and 3 listed inside of braces:
{1, 2, 3}
Alternatively we may choose to describe this set like this
{1, 2, 3} = { integers greater than 0 and less than 4 }
More formally, we may use this notation:
{1, 2, 3} = { x : x is an integer and 0 < x < 4} = { x : x  Z, 0 < x < 4 }
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IGCSE Mathematics – Sets and set notation
Singleton set
Singleton set is a set with only one element.
A singleton set is different from the element itself and it is denoted by the element with surrounding braces.
Ex.
5 – this is number 5
{5} –tihs is singleton set containing number 5
Empty Set
A set with no elements in it is called an empty set.
This set is unique and it is denoted with the symbol Ø or with empty braces { }.
Infinite set
Set which contain an infinite number of elements is called infinite set.
These sets are often given by a few elements and ellipses which indicate that this sequence continues
indefinitely.
A = {1, 2, 3, … }
Subsets
If all the elements of one set X are also elements of another set Y, then X is said to be a subset of Y. This is
written as
X Y
Similarly Z  Y implies that Z is not a subset of Y.
Superset
If a set X is a subset of Y, then Y is said to be a superset of X. This is written as
Y X
Every set is its own subset and superset.
The empty set is a subset of all sets.
Example 2
A A
A A
ØA
Let A = {1, 2, 3} and B = {1, 2}.
B is a subset of A
A is not a subset of B
B A
A B
Notice that: n (B) < n (A).
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IGCSE Mathematics – Sets and set notation
SET OPERATIONS
Example: Let A = {10, 11, 12, 13} and B = {12, 13, 14}.
Intersection
Symbol A ∩ B denotes the intersection of two sets.
It is a set comprised of the elements which are in the both sets. We can write this as:
A ∩ B = { x| x  A and x  B }
In the example A ∩ B = {12, 13}
If two sets have no elements in common, i.e. A ∩B = Ø we say that they are disjoint.
Union
Symbol A U B denotes the union of two sets.
It is a set comprised of the elements which are in the either set (at least one). We write this as:
A U B = { x : x  A or x  B }
In the example A U B = {10, 11, 12, 13, 14}
Universal set
The set containing all elements of a problem is called the universal set (ℰ).
That is a set from which elements may be considered.
(The notation comes from the French word ensemble).
Complement
If ℰ is universal set and A is a subset of ℰ, then the complement of a set A, denoted by A’, is the
set of all elements not in A.
A’ = { x : x  A }
Cardinality of sets (Number of elements)
The number of elements in a set A is called its cardinality and is denoted by n (A)
Example.
If A = {a, b, c, d, e, f} then n (A) = 6.
Properties of Cardinality:
1.
n(Ø) = 0
2.
3.
4.
n(A’) = n(ℰ) – n(A)
if A ∩ B ≠ Ø then
n(AUB) = n(A) + n(B) – n(A ∩ B)
if A ∩ B = Ø (disjoint sets) then
n(AUB) = n(A) + n(B)
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