Introduction to crisp set

Crisp set is a collection of unordered distinct elements, which are derived from Universal set. Universal set consists of all possible elements which take part in any experiment. Set is quite useful and important way of representing data.

Let X represents a set of natural numbers, so

X = {1, 2, 3, 4, …}

Sets are always defined with respect to some universal set. Let us derive two sets A and B from this universal set X.

A = Set of even numbers = {2, 4, 6, …}

B = Set of odd number = {1, 3, 5, …}

Elements in the set are unique, i.e. A = {1, 1, 2, 2, 3, 3}, B = {1, 2, 3}, C = {1, 2, 2, 3, 3, 3} all are same.

Order of elements in the set is not important, i.e. A = {1, 2, 3}, B = {2, 1, 3}, C = {3, 1, 2}, all corresponds to identical set.

Element of the set is called member of the set. If any element is present in the set then it is considered as a member of set otherwise it is not member. In crisp set, there is no concept of partial membership. Element is either fully present in the set or it is fully outside the set.

Crisp set is very important to model or to represent many real-world entities, such as set of boys, set of books, set of elements, set of employees, set of colors etc.

The membership function can be used to define a set A is given by

\chi_{A}(x) = \begin{cases} 1, & if x \in A \\ 0, & if x \notin A \end{cases}

The function χ ( read as ‘chi‘ ) is known as crisp membership function, which assigns membership value to the element of universal set based on certain properties.

Examples of crisp set

Let us discuss an example of crisp set. consider X represents class of students which acts as Universe of discourse. If you ask a question, “who does have a driving license?”  Obviously, all students might not have driving license. So those students who have driving license will have membership value 1 for this particular set and rest of all will have membership value zero.

We can define set A is equal to set of students having driving license and A will be definitely subset of universal set X

universal set
Universal set
Crisp set
Crisp Set

The best example of crisp set representation is number system in mathematics, where,

  • N: Set of natural numbers
  • R: Set of real numbers
  • Z: Set of integers
  • Q: Set of rational numbers

Notations used in Crisp Set

We will discuss the various set notations with respect to following sets:

X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10}
B = {1, 3, 5, 7, 9}
C = {4, 6, 8}
D = {x | x is perfect square and x > 10} = Φ

Various notations used in set theory are defined below:

  • Φ: Empty set is represented by the symbol, Φ is a set which does not have any element in it. For given data, D = Φ
  • x ∈ A represents element x is member of set A. For given data, 2 ∈ A
  • x ∉ A represents an element x is not a member of set A. For given data, 3 ∉ A
  • A ⊆ B represents every element of set A is present in set B as well. In other words, A is subset of B. For given data, A ⊆ X
  • A ⊇ B represents every element of B is member of set A as well. In other words, A is superset of B. For given sets, A ⊇ C
  • A ⊂ B represents every element of A is in B as well as B has some additional element which is not in A. This notation says that A is proper subset of B.
  • A ⊃ B represents all the elements of B are in set A as well as A has some additional element which is not in B. This notation says that A is proper superset of B.
  • if set A and B are identical then we can say A is subset of B or B is subset of A, but we cannot say that A is proper superset of B or A is proper subset of B
  • A = B represents Equal sets, i.e. sets A and B have identical elements
  • A ≠ B represents Not equal sets, i.e. sets A and B have different elements. For given sets, A ≠ B
  • |A| represents Cardinality of set A (i.e. number of elements in set A). For given sets, |A|= 5
  • p(A) represents Power set of set A. For the given sets, p(c) = { Φ, {4}, {6}, {8}, {4, 6}, {4, 8}, {6, 8}, {4, 6, 8} }
  • |p(A)|: 2|A|, i.e. power set of any set contains 2n elements.

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Crisp set is very convenient to represent and process the data. Many programing languages including Python, R etc. supports the creation and manipulation of sets.

Venn diagram representation of sets

Sets and set operations are often represented using Venn diagram, which is a graphical representation of interaction between sets. Let us try to understand it with example. Assume the use case of some university having 100 students. To form the team for inter college sport tournaments, interests of the students were collected and we got following responses. Out of 100 students,

  • 60 students were interested in football only
  • 50 students were interested in cricket only
  • 45 students were interested in volleyball only
  • 30 students were interested in football and cricket both
  • 50 students were interested in volleyball and cricket both
  • 20 students were interested in football and volleyball both
  • 5 students were interested in all three games

This fact can be easily and elegantly represented by Venn diagram as shown below, where each circle corresponds to one set. Their overlapping shows the intersection of respective sets. Graphical representation is easy to understand and quick to analyze compared to its counter textual representation.

Venn diagram representation for sets
Venn Diagram for sets

Suggested Reading: Operations on crisp set

Test Your Knowledge:

X = {1, 2, 3, 4, …, 20}

A = Set of perfect square

B = Set of odd numbers

C = Set of numbers which are divisible by 3

D = Set of numbers which are divisible by 6

For above given sets, answer the following:

  1. Find the pair of sets which are subset and superset of each other
  2. Is D subset of A
  3. Find the power set of set C
  4. Is set C subset of set B?
  5. Is A subset of C?

Please post your answers / queries in comment box below !