COMPUTATIONAL MATHEMATICS - SET THEORY

 

·         A set can be defined as a collection of things that are brought together because they obey a certain rule.

·         These 'things' may be anything you like: numbers, people, shapes, cities, bits of text ..., literally anything.

TYPES OF SETS

·         Universal Set - The set of all the 'things' currently under discussion is called the universal set (or sometimes, simply the universe). It is denoted by U.

·         Singleton Set: A set which contains only one element is called a singleton set.

For example:

• A = {x : x is neither prime nor composite} It is a singleton set containing one element, i.e., 1.

• B = {x : x is a whole number, x < 1} This set contains only one element 0 and is a singleton set.

• Let A = {x : x ∈ N and x² = 4} Here A is a singleton set because there is only one element 2 whose square is 4.

• Let B = {x : x is a even prime number} Here B is a singleton set because there is only one prime number which is even, i.e., 2.

·         Finite Set: A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.

For example:

• The set of all colors in the rainbow.

• N = {x : x ∈ N, x < 7}

• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}

Subset

Consider the sets, X = set of all students in your school and Y = set of all students in your class. It is obvious that set of all students in your class will be in your school. So, every element of Y is also an element of X. We say that Y is a subset of X. The fact that Y is a subset of X is expressed in symbol as Y⊂X. The symbol ⊂ stands for "is a subset of" or "is contained in".If Y is a subset of X, then X is known to be a superset of Y.

Complement of a Set

If we have a set A, then the set which is denoted by U - A, where U is the universal set is called the complement of A. Thus, it is the set of everything that does not belong to A. So, the complement of a set is the set of those elements which does not belong to the given set but belongs to the universal set U. Mathematically, we can show it as Ac = {x \ x ∉ A but x ∈ U}