Maths Set Theory

Set Theory

Mathematics - Set Theory
Written by Mary Christina

INTRODUCTION:

The concept of set is one of the fundamental concept in Mathematics. The notation and concept of set theory is so useful in every part of Mathematics.

Set Theory originated from from the works of George Boole and George Cantor.
It has helped in connecting many disconnected ideas and thus helped for the advancement in Mathematics.

SETS:

Definition:

A set is a collection of Well-defined objects. The objects in a set are called elements or members of that set.

Well-defined means decides whether the object belongs to that set or not and should be defined without any confusion.

Example:  The collection of ” bike users” in Chennai does not form a set, because in this case “bike users” is not well defined and thus this collection does not form a set.

Notation:

We generally represent set in capital letters like A,B,Y, etc. We use small letters like l,m,n, etc to represent the elements in the set.
x∈Y means x is an element of the set Y and if z∉Y means z is not an element of Y.

Examples:

  • The set of all employees working in Chennai
  • The set of all odd numbers
  • The set of all natural numbers
  • The set of all square numbers

DEFINITIONS:

Finite Set:

A set is said to be finite if it has finite number of elements

Infinite Set:

A set is said to be infinite if it does not contain finite number of elements.

Cardinality of a set:

If the set X is finite then the cardinality of the set is denoted by the number of elements present in the set X.
i.e Cardinality of the set is given by n(X)
If the set X is infinite then the cardinality of the set is denoted by the symbol ∞

Subset:

Let X and Y be two sets. We say X is a subset of Y if every element in X is also an element in Y i.e if z∈X then z∈Y also.
If X is a subset of Y then we denote X⊆Y

Note: Every set is a subset of itself.

Equal Sets:

Two sets X and Y are said to be equal if both the sets contain same elements.
In this case we write X=Y. That is X=Y if only if X⊆ Y and Y⊆ X.

Example:
Let P= {x/x is positive even numbers< 12} and Q={ 2,4,6,8,10}
Here the sets P and Q have exactly the same elements. Therefore P and Q are Equal sets.

Equivalent Sets:

Two sets X and Y are said to be equivalent if both the sets contain same number of elements i.e. n(X) = n(Y).

Example:
Let A= {a,e,i.o.u} and B={ 1,3,5,7,9}
Here the sets A and B contains same number of elements i.e. n(A)= 5 and n(B)=5..Therefore the sets A and B are Equivalent sets.

Power Sets:

Let X be a set then the power set P(X) denotes the collection of all the subsets of X.
P(X) is called the power set of X.
i.e. If n(X)= m Then the number of elements in P(X) is given by n(P(X))= 2m.

Example:
If the set X= { 1,2,3} Then the power set P(X) ={ { }, {1}, {2}, {3}, {1,2}, {2,3}, {3,1}, {1,2,3}}. and hence n(P(X))= 23=8.

About the author

Mary Christina

Mary Christina from Chennai, India. She has completed Master Degree in Mathematics and she loves to teach and share about her knowledge in maths for High School and College Students...

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