3. Definition

4. P - denoting the set of all primes.
N - denoting the set of all natural numbers.
Z - denoting the set of all integers.
z⁺- denoting the set of all positive integers.
Q - denoting the set of all rational numbers.
Q⁺- denoting the set of all positive rational numbers.
R - denoting the set of all real number.
R⁺- denoting the set of all positive real number.

5. Describing sets
There are two ways of describing, or specifying the
members of, a set.
1.)Roster or Tabular Form:In this form all the
element are listed, the elements are separated by
comas and are enclosed within braces { }.
Example : set of all positive natural number less than 11 is described in
roaster form as {1,2,3,4,5,6,7,8,9,10}
2.)Set Builder Form: In this form set, a set is
written in such a way that it follows rule or semantic
description
Example: {x: x ∈ N, 1 ≤ x ≤ 10}

6. FINITE SET
INFINITE SET
Finite set is a set having finite number
of sets.
Example: {1,2,3,4,5,6,7}
Infinite set is a set having endless number
of set. i.e. infinite number of sets.
Example: {1,2,3,4,5,6,7........}

7. EMPTY SET
SUBSET
Two sets A and B are said to be equal if they
Have exactly the same elements and we write
A=B. Otherwise, the are said to be unequal and
We write A≠B.
A set A is said to be subset of B if every
elements
of A is also an elements of B
Proper Subsets
A is a proper subset of B if and only if every
element in A is also in B, and there exists at least
one element in B that is not in A.

8. POWER SET
The collection of all subset of a set A is called the
power set of A. It is denoted by P(A). In P(A),
every element is a set.
Example: P(A)={1,2}, then
P(A)={φ,{1},{2},{1,2}}
UNIVERSAL SET
At the start we used the word "things" in quotes.
We call this the universal set. It's a set that
contains everything. Well, not exactly everything.
Everything that is relevant to our question. Then
our sets included integer.

9. VENN DIAGRAMS
•Represents sets graphically
•The box represents the universal set
•Circles represent the set(s)
•Consider set S, which is the set of all vowels in
the
alphabet
•The individual elements are usually not written
in a Venn diagram

10. 1.) Unions
The union of A and B is the set which contains of all
the elements of A and all the elements of B, the
common elements being taken only once. It is
denoted by ‘ ∪ ’
SET OPERATIONS
Formal definition for the union of
two sets:
A U B = { x | x  A or x  B }
A U B
A B
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

11. •A ∪ B = B ∪ A.
•A ∪ (B ∪ C) = (A ∪ B) ∪ C.
•A ⊆ (A ∪ B).
•A ∪ A = A.
•A ∪ ∅ = A.
•A ⊆ B if and only if A ∪ B =B.
Some basic properties of unions:

12. Intersection of Sets
The intersection of A and B, denoted by A ∩ B, is
the set of all things that are members of both A and
B. If A ∩ B = ∅, then A and B are said to be disjoint.
A B
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.
{1, 2} ∩ {2, 3} = {2}.

13. Some basic properties of intersections:
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B if and only if A ∩ B = A.

14. Complements of Sets
Two sets can also be "subtracted". The relative
complement of B in A (also called the set-theoretic
difference of A and B), denoted by A B (or A −
B), is the set of all elements that are members of
A but not members of B.
In certain settings all sets under discussion are
considered to be subsets of a given universal set U.
In such cases, U A is called the absolute
complement or simply complement of A, and is
denoted by A′.
Examples:
{1, 2} {1, 2} = ∅.
{1, 2, 3, 4} {1, 3} = {2, 4}.

15. The relative
complement
of B in A
The complement of A in U
The symmetric difference of A and B
A B
A B
A B

16. Some basic properties of complements:
•A ∪ A′ = U.
•A ∩ A′ = ∅.
•(A′)′ = A.
•A A = ∅.
•U′ = ∅
•∅′ = U.
•A B = A ∩ B′.

17. Note that:
• A = 
• A = 
• For non-empty sets A and B: AB  AB  BA
• |AB| = |A||B|
The Cartesian product of two or more sets is
defined as:
A1A2…An = {(a1, a2, …, an) | aiA for 1  i  n}
The Cartesian product of two sets is defined
as: AB = {(a, b) | aA ∩ bB}

18. De Morgan stated two laws about Sets.
If A and B are any two Sets then,
(A ∪ B)′ = A′ ∩ B′
The complement of A union B equals the
complement of A intersected with the
complement of B.
(A ∩ B)′ = A′ ∪ B′
The complement of A intersected with B is
equal to the complement of A union to the
complement of B.
De Morgan's Law