Sets Class 11 One Shot: Comprehensive Study Notes
This session provides a detailed, one-shot explanation of Sets for Class 11 Mathematics, covering all fundamental concepts, representations, types, operations, and proof techniques essential for school and competitive exams.
1. Introduction to Sets
Definition of a Set
A set is a well-defined collection of objects.
- Well-defined objects: Objects whose collection does not change from person to person. For example, "natural numbers less than 7" is a well-defined collection (1, 2, 3, 4, 5, 6), but "three most beautiful girls in your class" is not, as it varies from person to person.
Elements of a Set
- The objects within a set are called its elements or members.
- Elements are typically represented by lowercase letters.
- Sets are usually denoted by capital letters (e.g., A, B, C).
- Belongs to (∈): The symbol
∈ indicates that an element belongs to a set. For example, 1 ∈ A means 1 is an element of set A.
- Does not belong to (∉): The symbol
∉ indicates that an element does not belong to a set. For example, 7 ∉ A.
- Key Properties of Elements:
- Elements are not repeated: Each distinct element is listed only once. For example, the set of letters in "FOLLOW" is {F, O, L, W}, not {F, O, L, L, O, W}.
- Order is immaterial: The order in which elements are listed does not change the set. For example, {1, 2, 3} is the same as {3, 1, 2}.
2. Representation of Sets
Sets can be represented in two main forms:
2.1 Roster Form (or Tabular Form)
- Elements are listed within curly braces
{} and separated by commas.
- Example: The set of natural numbers less than 6 is
A = {1, 2, 3, 4, 5}.
2.2 Set-Builder Form
- Elements are described by a common property they all possess.
- Format:
{x : property of x} or {x | property of x}.
- Example: The set of natural numbers less than 6 is
A = {x : x is a natural number and x < 6}.
- Example: The set of all two-digit numbers which are perfect squares:
A = {x : x = n², where n ∈ N and 4 ≤ n ≤ 9}. This would result in {16, 25, 36, 49, 64, 81}.
3. Important Categories of Numbers as Sets
- Natural Numbers (N):
{1, 2, 3, ...}
- Whole Numbers (W):
{0, 1, 2, 3, ...}
- Integers (Z):
{..., -3, -2, -1, 0, 1, 2, 3, ...}
- Positive Integers (Z⁺):
{1, 2, 3, ...} (same as Natural Numbers)
- Negative Integers (Z⁻):
{..., -3, -2, -1}
- Rational Numbers (Q): Numbers that can be expressed as p/q, where p, q ∈ Z and q ≠ 0.
- Irrational Numbers (T): Numbers that cannot be expressed as p/q.
- Real Numbers (R): All rational and irrational numbers.
4. Types of Sets
4.1 Empty Set (Null Set or Void Set)
- A set containing no elements.
- Represented by
{} or Φ (phi).
- Example: The set of even prime numbers greater than 5 is
Φ.
- Important Note:
{Φ} is not an empty set; it is a singleton set containing the empty set as its element.
4.2 Singleton Set
- A set containing exactly one element.
- Example: The set of even prime numbers is
{2}.
4.3 Finite Set
- A set with a countable number of elements.
- The number of elements in a finite set is called its cardinal number or cardinality, denoted as
n(A).
- Example:
A = {1, 2, 3, 4, 5}, n(A) = 5.
- Note: The empty set
Φ is a finite set because n(Φ) = 0.
4.4 Infinite Set
- A set with an uncountable number of elements.
- Example: The set of natural numbers
{1, 2, 3, ...}.
- Important Note: Many infinite sets (like real numbers, rational numbers, irrational numbers) cannot be represented in Roster form due to the lack of a discernible pattern between consecutive elements.
4.5 Equivalent Sets
- Two sets A and B are equivalent if they have the same number of elements (same cardinality).
n(A) = n(B).- Example:
A = {1, 2, 3} and B = {a, b, c} are equivalent sets.
4.6 Equal Sets
- Two sets A and B are equal if they contain exactly the same elements.
- Example:
A = {1, 2, 3} and B = {3, 1, 2} are equal sets.
5. Subsets
Definition of a Subset
- Set A is a subset of set B if every element of A is also an element of B.
- Represented by
A ⊆ B.
- Example: If
A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
Proper Subset and Improper Subset
- Proper Subset (A ⊂ B): If A is a subset of B, and A is not equal to B (i.e., B contains at least one element not in A).
- Improper Subset: If A is a subset of B, and A is equal to B. Every set is an improper subset of itself.
- Super Set (B ⊃ A): If A is a subset of B, then B is a super set of A.
Key Properties of Subsets
- Every set is a subset of itself:
A ⊆ A.
- The empty set (Φ) is a subset of every set:
Φ ⊆ A.
Total Number of Subsets
- For a set with
n elements, the total number of subsets is 2^n.
- The total number of proper subsets is
2^n - 1 (excluding the set itself).
Power Set
- The power set of a set A, denoted as
P(A), is the collection of all possible subsets of A.
- Example: If
A = {1, 2}, its subsets are Φ, {1}, {2}, {1, 2}.
Then, P(A) = {Φ, {1}, {2}, {1, 2}}.
- The number of elements in the power set
n(P(A)) is 2^n(A).
6. Intervals as Subsets of Real Numbers
Intervals are used to represent infinite sets of real numbers that cannot be listed in Roster form.
- Closed Interval [a, b]:
{x : a ≤ x ≤ b, x ∈ R}. Includes both 'a' and 'b'.
- Open Interval (a, b):
{x : a < x < b, x ∈ R}. Excludes both 'a' and 'b'.
- Semi-Open/Semi-Closed Intervals:
(a, b]: {x : a < x ≤ b, x ∈ R}. Excludes 'a', includes 'b'.[a, b): {x : a ≤ x < b, x ∈ R}. Includes 'a', excludes 'b'.
- Length of an Interval: For any interval
[a, b], (a, b), [a, b), or (a, b], the length is b - a.
7. Universal Set
- The universal set, denoted by
U, is the largest set in a particular context or discussion, containing all elements relevant to that context.
- All other sets under consideration are subsets of the universal set.
- Example: If discussing natural numbers, the universal set could be
N, Z, Q, or R.
8. Operations on Sets
8.1 Union of Sets (A ∪ B)
- The union of two sets A and B is the set of all elements that are in A, or in B, or in both.
A ∪ B = {x : x ∈ A or x ∈ B}.- Properties:
- Commutative Law:
A ∪ B = B ∪ A
- Associative Law:
(A ∪ B) ∪ C = A ∪ (B ∪ C)
- Idempotent Law:
A ∪ A = A
- Law of Identity:
A ∪ Φ = A
- Law of Universal Set:
A ∪ U = U
8.2 Intersection of Sets (A ∩ B)
- The intersection of two sets A and B is the set of all elements that are common to both A and B.
A ∩ B = {x : x ∈ A and x ∈ B}.- Properties:
- Commutative Law:
A ∩ B = B ∩ A
- Associative Law:
(A ∩ B) ∩ C = A ∩ (B ∩ C)
- Idempotent Law:
A ∩ A = A
- Law of Identity:
A ∩ U = A
- Law of Empty Set:
A ∩ Φ = Φ
- Disjoint Sets: If
A ∩ B = Φ, then A and B are called disjoint sets (they have no common elements).
8.3 Difference of Sets (A - B)
- The difference of set A and set B is the set of elements that are in A but not in B.
A - B = {x : x ∈ A and x ∉ B}.- Note:
A - B ≠ B - A in general.
8.4 Symmetric Difference of Sets (A Δ B)
- The symmetric difference of two sets A and B is the set of elements that are in A or in B but not in their intersection.
A Δ B = (A - B) ∪ (B - A).
8.5 Complement of a Set (A')
- The complement of a set A (with respect to a universal set U) is the set of all elements in U that are not in A.
A' = U - A = {x : x ∈ U and x ∉ A}.- Properties:
- Complement Laws:
A ∪ A' = U and A ∩ A' = Φ
- Law of Double Complementation:
(A')' = A
- Laws of Empty Set and Universal Set:
Φ' = U and U' = Φ
8.6 De Morgan's Laws
(A ∪ B)' = A' ∩ B'(A ∩ B)' = A' ∪ B'
8.7 Distributive Laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
9. Proofs of Set Identities
To prove that two sets A and B are equal (A = B), you must show that:
A is a subset of B (A ⊆ B).B is a subset of A (B ⊆ A).
Step-by-step method for proving A ⊆ B:
- Assume an arbitrary element
x belongs to set A (Let x ∈ A).
- Using the definitions of set operations and given conditions, logically deduce that
x must also belong to set B (Therefore, x ∈ B).
- Conclude that
A ⊆ B.
Example Proof: Show that if A ⊆ B, then C - B ⊆ C - A.
- Assume:
x ∈ (C - B).
- By definition of difference:
x ∈ C and x ∉ B.
- Given:
A ⊆ B. This means if an element is not in B, it cannot be in A (because if it were in A, it would also have to be in B).
- Deduce: Since
x ∉ B and A ⊆ B, it implies x ∉ A.
- Combine: We have
x ∈ C and x ∉ A.
- By definition of difference:
x ∈ (C - A).
- Conclusion: Since
x ∈ (C - B) implies x ∈ (C - A), therefore C - B ⊆ C - A.
10. Practical Problems on Union and Intersection of Sets
These problems involve finding the number of elements in various combinations of sets, often in survey-like scenarios.
Formulas for Cardinality
- For two sets A and B:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- For three sets A, B, and C:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
Problem-Solving Method (using formulas):
- Identify the given cardinalities for individual sets and their intersections.
- Use the appropriate formula (for two or three sets) to find the cardinality of their union.
- If asked for elements not in the union (e.g., "neither A nor B"), subtract
n(A ∪ B) from the total number of elements in the universal set n(U).
Example: In a survey of 400 students, 100 took apple juice (A), 150 took orange juice (O), and 75 took both. Find how many took neither.
n(U) = 400, n(A) = 100, n(O) = 150, n(A ∩ O) = 75.n(A ∪ O) = n(A) + n(O) - n(A ∩ O) = 100 + 150 - 75 = 250 - 75 = 175.- Students taking neither =
n(U) - n(A ∪ O) = 400 - 175 = 225.
11. Questions for Reflection or Further Study
- How does the concept of "well-defined" distinguish a set from a mere collection of objects? Provide an example of a collection that is not a set and explain why.
- Explain the difference between a proper subset and an improper subset with an example.
- Why can't infinite sets like real numbers be represented in Roster form? How do intervals address this challenge?
- Consider De Morgan's Laws. Can you intuitively explain why
(A ∪ B)' = A' ∩ B'?
- In practical problems involving surveys, why is it important to subtract the intersection when calculating the union of two sets?