NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.5

NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 Sets focuses on one of the most interesting operations in set theory: the Complement of a Set. While previous exercises dealt with combining sets, this exercise explores what lies outside a specific set within a given Universal Set.

Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 1.5 introduces the concept of "not belonging to a set" and covers the famous De Morgan’s Laws. These laws are essential for simplifying complex logical expressions in both mathematics and computer science.

NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.5

  1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, and C = {3, 4, 5, 6}. Find:
  • i. A'
    • The complement of set A is the set of all elements of U which are not the elements of A.
    • A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4} = {5, 6, 7, 8, 9}.
  • ii. B'
    • B' = U - B = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8} = {1, 3, 5, 7, 9}.
  • iii. (A union C)'
    • First, find the union: A union C = {1, 2, 3, 4, 5, 6}.
    • (A union C)' = U - (A union C) = {7, 8, 9}.
  • iv. (A union B)'
    • A union B = {1, 2, 3, 4, 5, 6, 8}.
    • (A union B)' = U - (A union B) = {5, 7, 9}.
  • v. (A')'
    • (A')' = A = {1, 2, 3, 4}.
  • vi. (B - C)'
    • First, find the difference: B - C = {2, 8}.
    • (B - C)' = U - (B - C) = {1, 3, 4, 5, 6, 7, 9}.
  1. If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
  • i. A = {a, b, c} -> A' = {d, e, f, g, h}.
  • ii. B = {d, e, f, g} -> B' = {a, b, c, h}.
  • iii. C = {a, c, e, g} -> C' = {b, d, f, h}.
  • iv. D = {f, g, h, a} -> D' = {b, c, d, e}.
  1. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
  • i. {x: x is an even natural number}
    • Ans: {x: x is an odd natural number}.
  • ii. {x: x is an odd natural number}
    • Ans: {x: x is an even natural number}.
  • iii. {x: x is a prime number}
    • Ans: {x: x is a positive composite number and x = 1}.
  • iv. {x: x is a natural number divisible by 3 and 5}
    • Ans: {x: x is a natural number that is not divisible by 3 or 5}.
  • v. {x: x + 5 = 8}
    • Ans: {x: x is a natural number and x is not equal to 3}.
  • vi. {x: x >= 7}
    • Ans: {x: x is a natural number and x < 7}.
  1. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that:
  • i. (A union B)' = A' intersection B'
    • A union B = {2, 3, 4, 5, 6, 7, 8}.
    • (A union B)' = {1, 9}.
    • A' intersection B' = {1, 3, 5, 7, 9} intersection {1, 4, 6, 8, 9} = {1, 9}.
    • Verified.
  • ii. (A intersection B)' = A' union B'
    • A intersection B = {2}.
    • (A intersection B)' = {1, 3, 4, 5, 6, 7, 8, 9}.
    • A' union B' = {1, 3, 5, 7, 9} union {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}.
    • Verified.
  1. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60 degrees, what is A'?
  • Ans: A' is the set of all equilateral triangles.

Key Concepts in Exercise 1.5

To successfully navigate this exercise, you must master the following definitions and rules:

    • Complement of a Set ($A'$ or $A^c$): Let $U$ be the Universal Set and $A$ be a subset of $U$. The complement of $A$ is the set of all elements of $U$ which are not elements of $A$. Mathematically, $A' = U - A$.

  • Properties of Complement Sets:

    • Complement Laws: $A \cup A' = U$ and $A \cap A' = \phi$.

    • Law of Double Complementation: $(A')' = A$.

    • Laws of Empty Set and Universal Set: $\phi' = U$ and $U' = \phi$.

  • De Morgan’s Laws: These are the two most critical formulas in this exercise:

    1. $(A \cup B)' = A' \cap B'$ (The complement of the union is the intersection of the complements).

    2. $(A \cap B)' = A' \cup B'$ (The complement of the intersection is the union of the complements).

FAQs – Class 11 Maths Chapter 1 Exercise 1.5 Sets

Q1. What is the prerequisite for understanding Complement of a Set?

You must have a clear understanding of the Universal Set ($U$) and the Difference of Sets, as the complement is essentially the difference between the Universal Set and the set in question.

Q2. How do you represent a Complement in a Venn Diagram?

In a Venn diagram, the complement of set $A$ is represented by shading the entire region inside the Universal Set rectangle except for the circle representing set $A$.

Q3. Are De Morgan's Laws frequently asked in exams?

Yes. Proving De Morgan's Laws using specific sets or illustrating them via Venn diagrams is a very common exam question for Class 11 students.

Q4. What is the "Double Complement" property?

It states that if you take the complement of a set twice, you return to the original set. Think of it like a "double negative" in grammar—they cancel each other out.