Representation of Sets
* A set is a well defined collection of objects.
* A set is denoted with a capital letter.
* The elements of a set are denoted with small letters.
* Two ways of representing sets:
Roster form
Set-builder form
* Only distinct elements are written while representing a set in the roster form.
Types of Sets
* Non-empty Finite Set: If the number of elements in a set S is a natural number than S is said to be non-empty finite set.
* Empty Set: a set that does not contain any elements is called an empty set, null set or a void set.
* Infinite Set: A set that is empty or consists of a definite number of elements is called a finite set otherwise, it called an infinite set.
* Equal Sets: Two sets A and B are said to be equal if they have exactly the same elements, and we write A = B. Otherwise, the sets are said to be unequal, we write
A ≠ B.
Universal Set
* A set P is said to be a subset of set Q if every element of P is also an element of Q.
* Every set is a subset of itself.
* An empty set is a subset of every set.
* If any set P contains only one element, then P is called a singleton set.
* The collection of all subsets of a set A is called the power set of A.
Venn Diagrams
* The concept of Venn diagrams was introduced by an English mathematician, John Venn.
* A Venn diagram is a closed bounded figure, usually a circle or an ellipse, with the elements written within it.
Operations on Sets
* The union of two sets A and B is set C, which consists of all those elements that are either in A or in B.
* The intersection of two sets A and B is the set of all those elements that belong to both A and B.
* The difference of two sets taken in order is the set of elements of the first set that do not belong to the second set.
* Let U be the universal set, and A be a subset of U. then the compliment of A is the set of all elements of U that are not the elements of A.
Application of Sets
* The following relations can be used to solve practical problems:
n(A U B) = n(A) + n(B), A ∩ B = ɸ
n(A U B) = n(A) + n(B) – n(A ∩ B) A and B are finite sets
n(A U B U C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)