Type of Sets
Aug 08, 2022, 16:45 IST
Definition of set
A set is a well-defined collection of distinct objects. Well-defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally, sets are denoted by capital letters A, B, C, X, Y, Z etc.
Representation of a set
Usually, sets are represented in the following ways
1.ROASTER FORM OR TABULAR FORM
In this form, we list all the member of the set within braces (curly brackets) and separate these by commas. For example, the set of all even numbers less than 10 and greater than 0 in the roster form is written as: A = {2,4, 6,8}
2.SET BUILDER FORM OR RULE FORM
In this form, we write a variable (say x) representing any member of the set followed by a property satisfied by each member of the set. A = {x| x 5, x N} the symbol ‘|’ stands for the words” such that”.
Related Topics
Types of sets
1.NULL/ VOID/ EMPTY SET
A set which has no element is called the null set or empty set and is denoted by (phi). The number of elements of a set A is denoted as n (A) as it contains no element. For example, the set of all real numbers whose square is –1.
2. SINGLETON SET
A set containing only one element is called Singleton Set.
3. FINITE AND INFINITE SET
A set, which has finite numbers of elements, is called a finite set. Otherwise it is called an in finite set. For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.
4. UNION OF SETS
Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is defined in the below mentioned pdf .
5. INTERSECTION OF SETS
It is the set of all the elements, which are common to all the sets. The symbol used for intersection of sets is explained in below mentioned pdf.
6. DIFFERENCE OF SETS
The difference of set A to B denoted as A - B is the set of those elements that are in the set A but not in the set B i.e. explained in below mentioned pdf.
Example: If A = {a, b, c, d} and B = {b, c, e, f} then A-B = {a, d} and B-A = {e, f}.
Symmetric Difference of Two Sets:
For two sets A and B, symmetric difference of A and B is given by (A – B) (B – A) and is denoted by explained in below mentioned pdf.
7. SUBSET OF A SET
A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘explained in below mentioned pdf.Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A B. e.g If A = {a, b, c, d} and B = {b, c, d}. Then BA or equivalently AB (i.e A is a super set of B). Total number of subsets of a finite set containing n elements is 2n.
8. Equality of Two Sets:
Sets A and B are said to be equal if explained in below mentioned pdf.
9. DISJOINT SETS
If two sets A and B have no common elements i.e. if no element of A is in B and no element of B is in A, then A and B are said to be Disjoint Sets. Hence for Disjoint Sets A and B n Some More Results Regarding the Order of Finite Sets:
Let A, B and C be finite sets and U be the finite universal set, then Example are mentioned in the below image
10. UNIVERSAL SET
A non-empty set of which all the sets under consideration are subsets is called the universal set. In any application of set theory, all the sets under consideration will likely to be subsets of a fixed set called Universal Set. As name implies it is the set with collection of all the elements and usually denoted by ‘U’. e.g. (1) set of real numbers R is a universal set for the operations related to real numbers.
11. COMPLEMENTARY SET
The complement of a set A with respect to the Universal Set U is difference of U and A. Complement of set A is denoted by (or AC) (or A). Thus is the set of all the elements of the Universal Set which do not belong to the set A.
12. POWER SET
The set of all subsets of a given set A is called the power set A and is denoted by P (A). P (A) = {S: S A} For example, if A = {1, 2, 3}, then P(A) = { ,{1},{2},{3},{1},{1,2},{1,3},{2.3},{1,2,3}} Clearly, if A has n elements, then its power set P(A) contains exactly 2n elements.
What Is a Cartesian Product?
Cartesian product is that the product of any 2 sets, however this product is actually ordered i.e, the resultant set contains all possible and ordered pairs such that the primary element of the pair comes from the primary set and the second element comes from the second set. Since their order of appearance is very important, we call them first and second elements respectively.
