Crisp Sets Explained

A set is a well defined collection of objects. Here the word well defined means the object either belongs to or doesn’t belongs to the set. Observe the crispness in the example definitions below:

  • A = { Firefox, Internet Explorer, Chrome }
  • B = { Blogger, WordPress, TypePad }

VENN DIAGRAM is the pictorial representation to denote a set.
Universal Set E Venn Diagram
Here,

  • E is called universal set (or) universe of discourse.
  • A is a set contained in the universal set E.

Universe of Discourse

It is a set which contains all possible elements with similar characteristics with reference to a particular context. With the elements of this set, any number of sets can be formed.
The features of the elements in E can be discrete, countable integers or continuous valued quantities on real line.

Examples of Universal Set:

  • The clock speed of computer processor
  • The operating currents of an electric motor.
  • The operation temperature of heat pump.
  • The magnitude on Richter scale of an earthquake.

Membership of Elements

An element ‘X’ is said to be a member of set A if X is contained in A or in other words it belongs to A. ( X ε A )

Cardinality of a Set

The number of elements in a set is called its cardinality. Cardinality of a set A is denoted by n(A) or |A| or #A

Family of Sets

A set whose members are sets themselves is referred to as family of sets.

A = { (1, 3, 5), (2, 4, 6), (5, 10, 15) }
Here (1, 3, 5), (2, 4, 6), (5, 10, 15) are individual sets contained in set A.

Null Set or Empty Set

If a set ‘A’ has no elements in it or is empty then it called Null Set or Empty Set. The cardinality of a null set is zero.

n(A) = 0

Singleton Set

Any set with a single element contained in it is called singleton set. The cardinality of such a set it 1.

|A| = 1

Subset

Venn Diagram showing A subset BGiven two sets ‘A’ and ‘B’ defined over universal set ‘E’. ‘A’ is said be a subset of ‘B’, if ‘A’ is fully contained in ‘B’. It is represented by A⊂B. When ‘A’ is equivalent or subset of B, then it is represented by A⊆B.

Superset

Venn diagram showing P is a superset of Q Given two sets ‘P’ and ‘Q’ defined on ‘E’ universe of discourse. ‘P’ is said to be a superset of ‘Q’, if ever element in ‘Q’ is contained in ‘P’. It is represented by P ⊃ Q.

Powerset

Powerset of ‘A’ is the set of all possible subsets that are derivable from set A including null set. Powerset is represented by P(A). The cardinality of P(A) is 2|A|

For A = {2010, 2011} whose cardinality is |A| = 2
P(A)= { (2010), (2011), (2010, 2011), ( ) }

What is a Countable Set and Uncountable Set

Countable Set

S is a given set, if we can define a one-one mapping from S to the set of natural numbers N. Then S is called as a countable set.

For example :-

  1. Every finite set is a countable set.
  2. Set of integers is a countable set.
  3. Set of rational numbers is a countable set.

Uncountable Set

A set which is not countable which implies there cannot be a one-one mapping possible from the given set to set of natural numbers then that set is called an uncountable set.

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