Power Set

The collection of all subsets of a set \(X\) is called the power set of \(X\). It is denoted by \(P(X)\). In \(P(X)\), every element is a set.

if \(X = \left\{ {1,2,3} \right\}\), then
\(X\left( A \right) = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {1,3} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\}} \right\}\)
Also, note that \(n\left[ {X\left( A \right)} \right] = 8 = {2^3}\)

In general, if \(X\) is a set with \(n\left( X \right) = m\), then it can be shown that
\(n\left[ {P\left( A \right)} \right] = {2^m}\)
Universal Set
A Universal is the set of all elements under consideration, denoted by capital U.
Venn diagram
Venn diagrams were introduced in 1880 by John Venn (1834–1923).
These diagrams consist of rectangles and closed curves usually circles.
The universal set is represented usually by a rectangle and its subsets by circles.
Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically
represents the elements of the set, while the exterior represents elements that are
not members of the set. For instance, in a twoset Venn diagram, one circle may
represent the group of all wooden objects,
while another circle may represent the set of all tables
Operation on Sets
Union of Sets
The union of two sets \(A\) and \(B\) is the set \(C\) which consists of
all those elements which are either
in \(A\) or in \(B\) (including those which are in both). In symbols, we write.
\(A \cup B = \left\{ {x:x \in A{\rm{ or }}x \in B} \right\}\)
Venn Digram
Some Properties of the Operation of Union
 Commutative law : \(X \cup Y = Y \cup X\)
 Associative law : \(\left( {X \cup Y} \right) \cup Z = X \cup \left( {Y \cup Z} \right)\)
 Law of identity element, \(\phi \) is the identity of \( \cup \) : \(X \cup \phi = X\)
 Idempotent law : \(X \cup X = X\)
 Law of U : \(U \cup X = U\)
Intersection of Sets
The Intersection of two sets \(A\) and \(B\) is the set \(C\) which consists of all those elements which are present in both \(A\) and \(B\) .
In symbols, we write.
\(A \cap B = \left\{ {x:x \in A{\rm{ and }}x \in B} \right\}\)
Venn Digram
Some Properties of Operation of Intersection
 Commutative law : \(X \cap Y = Y \cap X\)
 Associative law : \(\left( {X \cap Y} \right) \cap Z = X \cap \left( {Y \cap Z} \right)\)
 Law of \( \cap \) and \(U\) : \(\phi \cap X = \phi \) , \(U \cap X = X\)
 Idempotent law : \(X \cap X = X\)
 Distributive law : \(X \cap \left( {Y \cup Z} \right) = \left( {X \cap Y} \right) \cup \left( {X \cap Z} \right)\)
Difference of set
The difference of two sets \(A\) and \(B\) is the set \(C\) which consists of all those elements which are
present in \(A\) but not in \(B\) . In symbols, we write,
\(A  B = \left\{ {x:x \in A{\rm{ and }}x \notin B} \right\}\)
Venn Digram
Some Properties of Operation of Difference

\(A  B \ne B  A\)

The sets \(\left( {A  B} \right)\) , \(\left( {A \cap B} \right)\) and \(\left( {B  A} \right)\) are mutually disjoint sets.
Compliment of set
Let \(U\) be the universal set and \(A\) a subset of \(U\). Then the complement of \(A\)
is the set of all elements of U\(U\) which are not the elements of \(A\).
Symbolically, we write \(A'\) to denote the complement of \(A\) with respect to \(U\).
Thus,
\(A' = \left\{ {x:x \in U{\rm{ and }}x \notin A} \right\}\) ,obviously \(A' = U  A\)
Venn Digram
Some Properties of compliment_of_sets
 Complement laws:
 \(A \cup A' = U\)
 \(A \cap A' = \phi \)
 De Morgan’s law:
 \(\left( {A \cup B} \right)' = A' \cap B'\)
 \(\left( {A \cap B} \right)' = A' \cap B'\)
 Law of double complementation: \(\left( {A'} \right)' = A\)
 Laws of empty set and universal set : \(\phi ' = U\) and \(U' = \phi \)
Cardinality of the set
 The cardinality of the set defines the number of element in the Set
 If \(A\) is the set, Cardinality of the set is defined as \(n(A)\)
 For \(A = \{ 1,2,3\} \) then \(n(A) = 3\)
Set Relations
Joined Set 
Disjoined Set 
Set having common elements 
Set having no common elements 
\(n\left( {X \cap Y} \right) \ne 0\) 
\(n\left( {X \cap Y} \right) = 0\) 
Important Operation on Cardinality

If \(n\left( {X \cap Y} \right) \ne 0\)
\(n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right)  n\left( {X \cap Y} \right)\)

If \(n\left( {X \cap Y} \right) = 0\)
\(n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right)\)
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