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Set theory and types of set in Discrete Mathematics

In this article, we will learn about the introduction of sets and the different types of set which is used in discrete mathematics.
Submitted by Prerana Jain, on August 11, 2018

Set theory

The set is a well-defined collection of definite objects of perception or thought and the Georg Cantor is the father of set theory. A set may also be thought of as grouping together of single objects into a whole. The objects should be distinct from each other and they should be distinguished from all those objects that do not from the set under consideration. Hence an st may be a bunch of grapes, a tea set or it may consist of geometrical points or straight lines.

A set is defined as an unordered collection of distinct elements of the same type where type is defined by the writer of the set.

Generally, a set is denoted by a capital symbol and the master or elements of a set are separated by an enclosed in { }.

    1 E A →   1 belong to A
    1 E/ A  → 1 does not belong to A

Types of set

There are many types of set in the set theory:

1. Singleton set

If a set contains only one element it is called to be a singleton set.

Hence the set given by {1}, {0}, {a} are all consisting of only one element and therefore are singleton sets.

2. Finite Set

A set consisting of a natural number of objects, i.e. in which number element is finite is said to be a finite set. Consider the sets

A = { 5, 7, 9, 11} and B = { 4 , 8 , 16, 32, 64, 128}

Obviously, A, B contain a finite number of elements, i.e. 4 objects in A and 6 in B. Thus they are finite sets.

3. Infinite set

If the number of elements in a set is finite, the set is said to be an infinite set.

Thus the set of all natural number is given by N = { 1, 2, 3, ...} is an infinite set. Similarly the set of all rational number between ) and 1 given by

A = {x:x E Q, 0 <x<1} is an infinite set.

4. Equal set

Two set A and B consisting of the same elements are said to be equal sets. In other words, if an element of the set A sets the set A and B are called equal i.e. A = B.

5. Null set/ empty set

A null set or an empty set is a valid set with no member.

A = { } / phie cardinality of A is 0.

There is two popular representation either empty curly braces { } or a special symbol phie. This A is a set which has null set inside it.

6. Subset

A subset A is said to be subset of B if every elements which belongs to A also belongs to B.

    A = { 1, 2, 3}
    B = { 1, 2, 3, 4}
    A subset of B.

7. Proper set

A set is said to be a proper subset of B if A is a subset of B, A is not equal to B or A is a subset of B but B contains at least one element which does not belong to A.

8. Improper set

Set A is called an improper subset of B if and Only if A = B. Every set is an improper subset of itself.

9. Power set

Power set of a set is defined as a set of every possible subset. If the cardinality of A is n than Cardinality of power set is 2^n as every element has two options either to belong to a subset or not.

10. Universal set

Any set which is a superset of all the sets under consideration is said to be universal set and is either denoted by omega or S or U.

    Let  A = {1, 2, 3}
    C = { 0, 1} then we can take
    S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.

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