CBSE Math CH 01: Sets Class -11th
Introduction to Sets
• A set is a well-defined collection of distinct objects.
➞ Sets are usually denoted by capital letters.
➞ The objects in a set are called elements or members.
Set of natural numbers less than 5, A = {1, 2, 3, 4}
⟹ If x is an element of a set, we write x ∈ A, which means x belongs to set A. Conversely, if x is not an element of set A, we write x ∉ A.
Representation of Sets
Sets can be represented in two ways:
- 1. Roster or Tabular Form
- 2. Set Builder Form
Roster or Tabular Form
⇒ In this form, all the members (elements) of the set are listed, the list is enclosed in curly brackets, and the elements are separated by commas.
B = {red, green, blue}
Set Builder Form
In this form, a set is defined by a property that its members must satisfy.
C = {x ; x is a positive integer less than 10}
Types of Sets
- Empty Set
- Singleton Set
- Finite Set
- Infinite Set
- Equal Sets
- Equivalent Sets
- Universal Set
- Subset
- Proper Subset
- Superset
- Proper Superset
- Power Set
1. Empty Set
A set which does not contain any element is called an empty set.
➡ It is denoted by { } or ∅.
➡ It is also known as Void set or Null set.
➥ Let A be the set of natural numbers divisible by 7 and less than 10. A = { }.
➥ D = {x ; x is a natural number less than 1} = ∅
2. Singleton Set
A set which contains only one element.
➥ Let B be the set of the number 5. B = {5}.
➥ I = {7}
3. Finite Set
A set which contains a definite number of elements is called a finite set.
➥ Let C be the set of vowels in the English alphabet. C = {a, e, i, o, u}.
➥ E = {2, 4, 6, 8}
4. Infinite Set
A set which contains an infinite number of elements is called an infinite set.
➥ Let D be the set of all real numbers. D = ℝ.
➥ F = {all natural numbers}
5. Equal Sets
Two sets are said to be equal if they contain exactly the same elements.
➥ Let E = {1, 2, 3} and F = {3, 2, 1}. E = F.
➥ G = {1, 2, 3} and H = {3, 2, 1} are equal sets.
6. Equivalent Sets
Two sets are said to be equivalent if they have the same number of elements.
➥ Let G = {a, b, c} and H = {1, 2, 3}. Both sets have 3 elements, so they are equivalent.
➥ A = {1, 2, 3} and B = {a, b, c} are equivalent sets.
7. Universal Set
The set that contains all the objects under consideration, usually denoted by U.
➥ Let U be the set of all students in a school. U = {all students}.
➥ If A={1,2,3}, B={2, 3, 4, 5} & C={2,3,5,6} then U (Universal set) = {1,2,3,4,5,6}
8. Subset
A set A is a subset of set B if all elements of A are also elements of B, denoted by A ⊆ B.
➥ Let I = {1, 2} and J = {1, 2, 3}. I ⊆ J.
➥ K = {1, 2} and L = {1, 2, 3}, then K ⊆ L.
9. Proper Subset
A set A is a proper subset of set B if all elements of A are in B and B contains at least one element not in A, denoted by A ⊂ B.
➥ Let K = {1, 2} and L = {1, 2, 3}. K ⊂ L.
➥ M = {1, 2} and N = {1, 2, 3}, then M ⊂ N.
10. Superset
If A is a subset of B, then B is a superset of A, denoted by B ⊇ A.
➥ Let M = {1, 2, 3} and N = {1, 2}. M ⊇ N.
➥ N = {1, 2, 3} and O = {1, 2}, then N ⊇ O.
11. Proper Superset
If A is a proper subset of B, then B is a proper superset of A, denoted by B ⊃ A.
➥ Let O = {1, 2, 3} and P = {1, 2}. O ⊃ P.
➥ P = {1, 2, 3} and Q = {1, 2}, then P ⊃ Q.
12. Power Set
The power set of a set A is the set of all subsets of A, including the empty set and A itself, denoted by P(A).
➥ Let Q = {1, 2}. The power set of Q is P(Q) = {{}, {1}, {2}, {1, 2}}.
➥ If M = {a, b}, then P(M) = {∅, {a}, {b}, {a, b}}
➥ If A = {1, 2, 3}, then P(A) = { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
Note: -
To find the number of subsets of a set A with n elements, you can use the formula:
Number of subsets = 2n
Where n is the number of elements in set A.
In your example, set A has 3 elements (1, 2, 3)
so, Number of subsets = 23 = 8
So, A has 8 subsets.
Set Builder Method: -
An elment many types of write of sets.
Let V = {a, e, i, o, u}.
➥so, V = {x: x vowel of english Alphabet}
➥ Or, V = {x: x in equation of vowel}
➥ An empty set has no proper subsets.
➥- Ek finit set jiska m elements hote hain, uske subsets ki sankhya 2m hoti hai.
- Proper subsets ki sankhya, yaani ki woh subsets jo original set ke samaan nahi hote, woh 2m - 1 hoti hai. Ye isliye kyunki ek set ke saare subsets mein se ek subset original set khud hoti hai, aur baaki 2m - 1 proper subsets hote hain.
Venn Diagrams
Venn diagrams are a way of picturing sets and their relationships to one another using circles.
Universal set (U) ⟹ Reactangle
and, Sets are represented by Circle
Que. A, B and C are subsets of Universal Set If A = {2, 4, 6, 8, 12, 20}, B= {3,6,9,12,15}, C= {5,10,15,20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.
Complement of a set
➥ If U an universal set an set A subset of U then complement of set A.
➥ It is denoted by [A'] = U-A
Let U = {1,2,3,4,5,6}
and, A = {1,2,4}
A' = U - A
A' = {1,2,3,4,5,6} - {1,2,4}
Hence, A'= {3,5,6}
⇨ Properties of Complement Sets
1. Complement law:
Ac = U - A
2. De Morgan's Law:
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
3. Bi-Complement Law:
(Ac)c = A
4. ∅ & U' Law:
A ∩ Uc = ∅
Note:
The above laws are fundamental in set theory and are widely used in various branches of mathematics and computer science.
Operations on Sets
1. Union of Two Sets
Definition: The union of two sets A and B is the set of elements that are in A, in B, or in both. It is denoted by A ∪ B.
Let A = {1, 2, 3} and B = {3, 4, 5}.
A ∪ B = {1, 2, 3, 4, 5}
A ∪ B = {1, 2, 3, 4, 6}
2. Intersection of Sets
Definition: The intersection of two sets A and B is the set of elements that are common to both A and B. It is denoted by A ∩ B.
Let A = {1, 2, 3} and B = {3, 4, 5}.
A ∩ B = {3}
A ∩ B = {2, 4}
3. Disjoint Sets
Definition: Two sets A and B are said to be disjoint if they have no elements in common. In other words, A ∩ B = ∅.
Let A = {1, 2, 3} and B = {4, 5, 6}.
A ∩ B = ∅
Hence, A and B are disjoint sets.
A ∩ B = ∅
Yes, they are disjoint sets.
4. Difference of Sets
Definition: The difference of two sets A and B, denoted by A - B, is the set of elements that are in A but not in B.
Let A = {1, 2, 3} and B = {3, 4, 5}.
A - B = {1, 2}
A - B = {6}
5. Symmetric Difference of Two Sets
Definition: The symmetric difference of two sets A and B, denoted by A Δ B, is the set of elements that are in either of the sets but not in their intersection.
Let A = {1, 2, 3} and B = {3, 4, 5}.
A Δ B = (A - B) ∪ (B - A)
A Δ B = {1, 2} ∪ {4, 5}
A Δ B = {1, 2, 4, 5}
A Δ B = (A - B) ∪ (B - A)
A Δ B = {6} ∪ {1, 3}
A Δ B = {1, 3, 6}
Algebra of Sets
1. Commutative Property
A ∪ B = B ∪ A and A ∩ B = B ∩ A
If A = {1, 2} and B = {2, 3}, then A ∪ B = B ∪ A = {1, 2, 3}
2. Associative Property
(A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C)
If A = {1}, B = {1, 2}, and C = {2, 3}, then (A ∪ B) ∪ C = A ∪ (B ∪ C) = {1, 2, 3}
3. Distributive Property
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
If A = {1}, B = {1, 2}, and C = {2, 3}, then A ∪ (B ∩ C) = {1, 2}
4. Identity Laws
A ∪ ∅ = A and A ∩ U = A
If A = {1, 2}, then A ∪ ∅ = {1, 2} and A ∩ U = {1, 2} (assuming U = {1, 2, 3})
5. Idempotent Laws
A ∪ A = A and A ∩ A = A
If A = {1, 2}, then A ∪ A = {1, 2}
Some Important Rules
-
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
where,- n(A ∪ B) = Number of elements in A ∪ B
- n(A) = Number of elements in Set A
- n(B) = Number of elements in Set B
- n(A ∩ B) = Number of elements in A ∩ B
- If A ∩ B = ∅, then n(A ∪ B) = n(A) + n(B)
- n(A - B) + n(A ∩ B) = n(A)
- n(B - A) + n(A ∩ B) = n(B)
- n(A ∪ B) = n(A - B) + n(A ∩ B) + n(B - A)
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)
- n(A \text{ only}) = n(A) - [n(A ∩ B) + n(A ∩ C) - n(A ∩ B ∩ C)]
- n(B \text{ only}) = n(B) - [n(A ∩ B) + n(B ∩ C) - n(A ∩ B ∩ C)]
- \text{Empty} = n(U) - [n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)]
- No. of elements in exactly two of the sets A, B, C:
n(A ∩ B) + n(B ∩ C) + n(C ∩ A) - 3n(A ∩ B ∩ C) - No. of elements in exactly one of the sets A, B, C:
n(A) + n(B) + n(C) - 2n(A ∩ B) - 2n(B ∩ C) - 2n(A ∩ C) - 3n(A ∩ B ∩ C)
