Set theory is the foundation of probability and statistics, as it is for almost every branch of mathematics.
Set is simply a collection of objects; the objects are referred to as elements of the set. The statement that x is an element of set S is written x ∈ S, and the negation that x is not an element of S is written as x ∉ S. By definition, a set is completely determined by its elements; thus sets A and B are equal if they have the same elements:
A = B if and only if x ∈ A ⟺ x ∈ B
If A and B are sets then A is a subset of B if every element of A is also an element of B:
A ⊆ B if and only if x ∈ A ⟹ x ∈ B
Every object in a set is unique: The same object cannot be
included in the set more than once.
Example 1:
Define each of the following sets by listing its
elements:
(a) The set of the first three months of the
year.
(b)
The sets of the whole numbers less than 2.
(c) The set of letters of the word ‘PAPER’.
Answer:
(a) {January,February,March}
(b)
{0,1}
(c) {P,A,E,R}
*Note on the Question C, we do not repeat the letter P and the sets has 4 element not 5.*
Example 2:
This question is on subsets.
I is the set of integers.
N is the set of natural numbers.
Q is the rational numbers
R is the set of real numbers.
(a) Use the symbol ⊂ to
relate I,N,Q,R.
(b) Draw a venn diagram to represent I,N,Q and R.
Answer:
(a) N⊂I⊂Q⊂R
(b)
Example 3:
Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}
a) Draw a Venn diagram to illustrate ( X ∪ Y ) ’
b) Find ( X ∪ Y ) ’
Solution:
a) First, fill in the elements for X ∩ Y = {1}
Fill in the other elements for X and Y and for U
Shade the region outside X ∪ Y to indicate (X ∪ Y ) ’
b) We can see from the Venn diagram that
(X ∪ Y ) ’ = {9}
Or we find that X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8} and so
(X ∪ Y ) ’ = {9} Video:
