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Sets




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 two-set 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

  1. Complement laws:
    • \(A \cup A' = U\)
    • \(A \cap A' = \phi \)
  2. De Morgan’s law:
    • \(\left( {A \cup B} \right)' = A' \cap B'\)
    • \(\left( {A \cap B} \right)' = A' \cap B'\)
  3. Law of double complementation: \(\left( {A'} \right)' = A\)
  4. 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

  1. If \(n\left( {X \cap Y} \right) \ne 0\)
    \(n\left( {X \cup Y} \right) = n\left( X \right) + n\left( Y \right) - n\left( {X \cap Y} \right)\)
  2. 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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