Concept: Sets
CONTENTS
sets are a collection of well-defined objects or elements. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}. For example, set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}.
Sets can be represented in 2 forms:
- Roster Form: All the elements of the set are listed. Example, the set of all odd natural numbers less than 11 is represented as {1, 3, 5, 7, 9}
- Set Builder Form: Set is described by a characteristising property. Example the set of all odd natural numbers less than 11 is represented as {x ∈ N | x < 12 and x is even}. | means 'such that'
Empty or Null Set: It has no element present in it. Example: A = {} is a null set.
Finite Set: It has a zero or limited number of elements. Example: A = {1,2,3,4}
Infinite Set: It has an infinite number of elements.Example: A = {x: x is the set of all whole numbers}
Equal Set: Two sets which have the same members.Example: A = {1,2,5} and B={2,5,1}: Set A = Set B
Subset Set: A set ‘A’ is said to be a subset of B if each element of A is also an element of B. Example: A = {1,2}, B = {1,2,3,4}, then A ⊆ B
- Proper subset: B. If A is a subset of B and there is at least one element in B that is not there in A, A is said to be a proper subset of B. This is written as A ⊂ B.
- A ⊄ B means A is not a subset of B.
- A ⊄ B means A is a not a proper subset of B.
- If A is a subset of B we say that B contains A or B is a superset of A. This is written as B ⊇ A
Note: Note: A ⊂ B ⇒ A ⊆ B. But the converse is not true.
Power Set: : If A is any set, then the set of all subsets of A is called the power set of A and is denoted by P(A), i.e., P(A) = {S│S ⊆ A}
Example: If A = {1, 2, 3}, then P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
Universal Set: The set that contains all the elements in a given context is called the Universal Set. It is denoted by µ or U.
: If A and B are two sets, the union of A and B is the set of all those elements which belong to either A or to B or to both A and B. This is denoted by A ∪ B.
A ∪ B = {x │x ∈ A or x ∈ B}
- If A ⊆ B, then A ∪ B = B
- A ∪ φ = A
- A ∪ µ = µ
Example: Given A = {1, 2, 3, 4, 5} while B = {4, 5, 6, 7, 8}. Find A ∪ B
Solution:
A ∪ B represents all the elements which belong to either A or to B or to both A and B.
∴ A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B. It is denoted by A ∩ B.
A ∩ B = {x | x ∈ A and x ∈ B}
- A ⊆ B, then A ∩ B = A
- A ∩ φ = φ
- A ∩ µ = A
Example: Given A = {1, 2, 3, 4, 5} while B = {4, 5, 6, 7, 8}. Find A ∩ B
Solution:
A ∩ B represents all the elements which belong to both A and B.
∴ A ∩ B = {4, 5}
Two sets A and B are said to be disjoint if they have no element in common.
∴ If A and B are disjoint, then A ∩ B = φ.
Example: A = {1, 3, 5} and B = {2, 4, 6}
Here, set A and B are disjoint sets since they have no common element.
The complement of set A is the set of all those elements that do not belong to set A, i.e. the complement of a set A is the difference of the universal set and set A and is denoted by A' or Ac.
Example: If µ is the set of natural numbers, A is the set of even natural numbers, then A' is the set of odd natural numbers.
Example: Given µ is the set of all natural numbers less than or equal to 10, while set A = {1, 2, 3, 4, 5}. Find A'
Solution:
µ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 2, 3, 4, 5}
∴ A' = {6, 7, 8, 9, 10}
The difference of two sets is the set of all elements which are there in one set but not in the other. Let A and B be two sets. A − B is the set of all those elements of A which do not belong to B.
A − B = {x | x ∈ A and x ∉ B}
Similarly, B − A = { x | x ∈ B and x ∉ A}.
Example: A = {1, 2, 3, 4}, B = {3, 4, 8, 10}
A − B = {1, 2}
B − A = {8, 10}