Class 11 — Mathematics (NCERT)

Sets: Operations and Venn Diagrams

Class 11

✓By the end of this lesson students will be able to define and perform various set operations such as union, intersection, difference, and complement.
✓By the end of this lesson students will be able to represent set operations and relationships using Venn diagrams.
✓By the end of this lesson students will be able to apply properties of set operations, including De Morgan's Laws, to simplify expressions and verify identities.
✓By the end of this lesson students will be able to solve practical problems involving sets using set operations and Venn diagrams.

Key concepts

Union of Sets

The union of two sets A and B, denoted by A ∪ B, is the set of all elements which are either in A or in B or in both. In a Venn diagram, A ∪ B is represented by the combined shaded regions of A and B.

A ∪ B = {x : x ∈ A or x ∈ B}

Intersection of Sets

The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are common to both A and B. In a Venn diagram, A ∩ B is represented by the overlapping region of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

Disjoint Sets

Two sets A and B are said to be disjoint if their intersection is the empty set, i.e., A ∩ B = ∅. In a Venn diagram, disjoint sets are represented by non-overlapping circles.

Difference of Sets

The difference of two sets A and B, denoted by A - B, is the set of elements which are in A but not in B. Similarly, B - A is the set of elements which are in B but not in A. In a Venn diagram, A - B is the part of circle A that does not overlap with circle B.

A - B = {x : x ∈ A and x ∉ B}

Complement of a Set

Let U be the universal set and A be any subset of U. Then the complement of A, denoted by A' or Aᶜ, is the set of all elements of U which are not in A. In a Venn diagram, A' is the region inside the universal set rectangle but outside circle A.

A' = {x : x ∈ U and x ∉ A}

Venn Diagrams

Venn diagrams are graphical representations of sets and their relationships. A universal set is represented by a rectangle, and its subsets are represented by circles within the rectangle. They are useful for visualising set operations and solving problems.

Properties of Set Operations

Set operations obey several fundamental laws:

Commutative Laws:\nA ∪ B = B ∪ A\nA ∩ B = B ∩ A\n\nAssociative Laws:\n(A ∪ B) ∪ C = A ∪ (B ∪ C)\n(A ∩ B) ∩ C = A ∩ (B ∩ C)\n\nDistributive Laws:\nA ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)\nA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)\n\nDe Morgan's Laws:\n(A ∪ B)' = A' ∩ B'\n(A ∩ B)' = A' ∪ B'\n\nIdentity Laws:\nA ∪ ∅ = A\nA ∩ U = A\n\nIdempotent Laws:\nA ∪ A = A\nA ∩ A = A\n\nComplement Laws:\nA ∪ A' = U\nA ∩ A' = ∅\n(A')' = A\n∅' = U\nU' = ∅

Key facts to remember

1A ∪ B = {x : x ∈ A or x ∈ B}
2A ∩ B = {x : x ∈ A and x ∈ B}
3A - B = {x : x ∈ A and x ∉ B} = A ∩ B'
4A' = {x : x ∈ U and x ∉ A} = U - A
5De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
6n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
7If A and B are disjoint sets, then A ∩ B = ∅ and n(A ∪ B) = n(A) + n(B).

Worked examples

Example 1

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7}. Find A ∪ B, A ∩ B, A - B, B - A, and A'.

ITo find A ∪ B, we combine all elements from A and B, taking common elements only once.

IIA ∪ B = {1, 2, 3, 4, 5, 6, 7}

IIITo find A ∩ B, we identify elements common to both A and B.

IVA ∩ B = {4, 5}

VTo find A - B, we list elements that are in A but not in B.

VIA - B = {1, 2, 3}

VIITo find B - A, we list elements that are in B but not in A.

VIIIB - A = {6, 7}

9To find A', we list elements in the universal set U that are not in A.

10A' = U - A = {6, 7, 8, 9, 10}

Answer

A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A ∩ B = {4, 5}, A - B = {1, 2, 3}, B - A = {6, 7}, A' = {6, 7, 8, 9, 10}

Example 2

Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. Verify De Morgan's First Law: (A ∪ B)' = A' ∩ B'.

IFirst, we find the L.H.S.: (A ∪ B)'.

IICalculate A ∪ B: A ∪ B = {1, 2, 3, 4, 5, 6}.

IIICalculate (A ∪ B)': (A ∪ B)' = U - (A ∪ B) = {1, 2, 3, 4, 5, 6, 7, 8} - {1, 2, 3, 4, 5, 6} = {7, 8}. (L.H.S.)

IVNext, we find the R.H.S.: A' ∩ B'.

VCalculate A': A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8} - {1, 2, 3, 4} = {5, 6, 7, 8}.

VICalculate B': B' = U - B = {1, 2, 3, 4, 5, 6, 7, 8} - {3, 4, 5, 6} = {1, 2, 7, 8}.

VIICalculate A' ∩ B': A' ∩ B' = {5, 6, 7, 8} ∩ {1, 2, 7, 8} = {7, 8}. (R.H.S.)

VIIISince L.H.S. = {7, 8} and R.H.S. = {7, 8}, we have L.H.S. = R.H.S. Hence proved.

Answer

(A ∪ B)' = {7, 8} and A' ∩ B' = {7, 8}. Thus, (A ∪ B)' = A' ∩ B' is verified.

Always calculate L.H.S. and R.H.S. independently and then compare for verification problems.

Example 3

In a group of 60 students, 25 students play cricket, 30 students play football, and 10 students play both cricket and football. Find the number of students who play at least one of the two games.

ILet C be the set of students who play cricket and F be the set of students who play football.

IIGiven: Number of students who play cricket, n(C) = 25.

IIIGiven: Number of students who play football, n(F) = 30.

IVGiven: Number of students who play both cricket and football, n(C ∩ F) = 10.

VWe need to find the number of students who play at least one of the two games, which is n(C ∪ F).

VIUsing the formula for the union of two sets: n(C ∪ F) = n(C) + n(F) - n(C ∩ F).

VIISubstitute the given values into the formula: n(C ∪ F) = 25 + 30 - 10.

VIIICalculate the result: n(C ∪ F) = 55 - 10 = 45.

Answer

45 students play at least one of the two games.

This problem can be effectively visualised using a Venn diagram to understand the overlap.

Common mistakes

✗Confusing the symbols and definitions of union (∪) and intersection (∩).
✗Incorrectly applying De Morgan's Laws, especially forgetting to change the operation (∪ to ∩ or ∩ to ∪) when taking the complement.
✗Forgetting to consider the universal set (U) when calculating the complement of a set.
✗Assuming A - B is the same as B - A; these are generally different sets.
✗Errors in counting elements for n(A ∪ B) by double-counting elements in the intersection.

Exam tips

★Always draw Venn diagrams for problems involving 2 or 3 sets to visualise the relationships and operations, especially for word problems.
★Memorise all the properties of set operations, particularly De Morgan's Laws, as they are frequently tested in various forms.
★For verification problems, always calculate the L.H.S. and R.H.S. separately and then compare the final sets to prove equality.
★Pay close attention to the universal set (U) provided in the question, as it is crucial for determining complements.

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