Concept Of Sets
It can be said that, set is to be considered as a collection of well defined objects. Every object of any collection forming a set is called “Member” or “Element” of the Set. When an object is a member of a set, then we can say object belongs to the sets. We can say that, any collection of objects is not a set.
Take The Collection of Object is Given Below -
1) The first four natural numbers - The members of the collection is 1, 2, 3, and 4. These objects are well defined. We can conclude that, we may in a position to decide whether a particular object belongs to the collection or not. 3 is a number of collections where 5 is not. Here we can conclude that, the numbers of the collection are distinct or different from each other and the collection is set.
2) Four consecutive natural numbers – It is not a Set. We are unable to decide whether an object belongs to the collection or not, for example, 4 may belongs to the collection or it may not. All the earlier said same number is true for any of the other natural numbers.
3) Collection of letters of the word COLLECTION.– It is not a set, because ‘L’ occurs in it twice, and hence, the objects of the collection are not distinct. However, if we take only one ‘L’ in place of two ‘L’, we get the collection C, O, L, E, C, T, I, O, N which is a set.
Sets are denoted by capital letters, like A, B, C, D, E,……. and the members or elements are denoted by as small letters, such as a, b, c, d, e, f,……… If ‘d’ is a member of Set ‘D’, then we can write as - “d ϵ D“ which should be read as ‘d’ belongs to ‘D’. If ‘d’ is not a member of set ‘D’ then we can write d ∉ D, d is not belongs to D.
Let us assume that, Z is the set of the first 4 natural numbers. Then 3 ∉ Z, 5 ∉ Z.
But, 2 ϵ Z, 4 ϵ Z.
1) This is to be considered that, the set of natural numbers is denoted by ‘N’. If 4 is a natural number then, 4 ϵ N, but -3 is not a natural number, so -3 ∉ N.
2) This is to be considered that, the set of whole numbers is denoted by ‘W’, where 0 is a whole number, so 0 ϵ W, but 1/7 is not a whole number, so 1/7 ∉ W.
3) The set of Integers are denoted by ‘Y’, if -8 is an Integer then -8 ϵ Y but 3/8 is not an Integer.
so, 3/8 ∉ Y.
4) The members of a set can be listed in any order. Thus the set { A, B, C, D } can be written as { D, C, B, A }
Some Important Symbol Of Sets –
SYMBOL MEANING
X ϵ A X is the member of the Set ‘A’
X ∉ A X is not a member of the set A
n(A) The number of distinct member of A
A ↔ B A & B are equivalent set
A = B A & B are equal sets, that is B ⊆ A, A ⊆ B
A ⊆ B A is a subset of B, that is every member of A is a member of B
n(φ) φ is the empty set
{ 0,1, 2, 3,…..} Set of whole numbers
{ 1, 2, 3, 4,…. } Set of natural numbers
{…,-3, -2, -1, 0, 1, 2,….} Set of Integers
Presentation of Set –
Presentation of Set can be in two ways -
1) Roster method or Tabular Method
2) Rule method or Set Builder method
Types of Set - 1) Finite Set
2) Infinite Set