Maths Project
SETS
Set and its notation
. Introduction : A set is an unordered collection of elements.
Examples.
{1, 2, 3} is the set containing “1” and “2” and
“3.”
{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is
irrelevant.
{1, 2, 3} = {3, 2, 1} since sets are unordered.
{0,1, 2, 3, …} is a way we denote an infinite set
(in this case, the natural numbers).
 = {} is the empty set, or the set containing no
element.
Note:   {}
Subsets
x  S means “x is an element of set S.”
x  S means “x is not an element of set S.”
A  B means “A is a subset of B.”
or, “B contains A.”
or, “every element of A is also in B.”
or, x ((x  A)  (x  B)).
A
Venn Diagram
B
Superset
A  B means “A is a subset of B.”
A  B means “A is a superset of B.”
A = B if and only if A and B have exactly the same elements
iff, A  B and B  A
iff, A  B and A  B
iff, x ((x  A)  (x  B)).
So to show equality of sets A and B, show:
AB
BA
A  B means “A is a proper subset of B.”
A  B, and A  B.
x ((x  A)  (x  B))
 x ((x  B)  (x  A))
A
B
Power sets
If S is a set, then the power set of S is
P(S) = 2S = { x : x  S }.
We say, “P(S) is the set of all
subsets of S.”
2S = {, {a}}.
If S = {a}
If S = {a,b}
If S = 
2S = {}.
If S = {,{}}
2S = {, {a}, {b},
{a,b}}.
2S = {, {}, {{}}, {,{}}}.
Union
The union of two sets A and B is:
A  B = { x : x  A  x  B}
If A = {Charlie, Lucy, Linus}, and
B = {Lucy, Desi}, then
A  B = {Charlie, Lucy, Linus, Desi}
B
A
Intersection
The intersection of two sets A and B is:
A  B = { x : x  A  x  B}
If A = {Charlie, Lucy, Linus}, and
B = {Lucy, Desi}, then
A  B = {Lucy}
B
A
Complement
The complement of a set A is:
If A = {x : x is not shaded}, then
U
A
= U
and
U=
Difference
The symmetric difference, A  B, is:
A  B = { x : (x  A  x  B)  (x  B  x  A)}
= (A – B)  (B – A)
= { x : x  A  x  B}
U
A–B
B–A
 Properties
of the union operation:
AU=A
Identity law
 AUU=U
Domination law
 AUA=A
Idempotent law
 AUB=BUA
Commutative law
 A U (B U C) = (A U B) U C
Associative law

 Properties
of the intersection
operation:
A∩U=A
Identity law
 A∩=
Domination law
 A∩A=A
Idempotent law
 A∩B=B∩A
Commutative law
 A ∩ (B ∩ C) = (A ∩ B) ∩ C
Associative law

Some Properties of Complement
Sets
1. Complement Laws:
A U A’= U
A ∩ A’= 
2. De Morgan’s Law:
(A U B)’ = A’ ∩ B’
(A ∩ B )’ = A’ U B’
3. Law of Double Complementation:
( A’ )’ = A
4. Laws of Empty Set and Universal Set:
’=U
U’ = 
THANK YOU