CBSE Class 10 Mathematics Sets Notes Set 16

Sets and Their Representation

A set is a well-defined collection of objects, called its elements. It can be written in roster form {1, 2, 3} or set-builder form {x : x is a natural number less than 4}. The empty set φ has no elements. Sets with a countable number of elements are finite; otherwise they are infinite. Two sets are equal if they have the same elements.

Subsets and Power Set

A is a subset of B (A ⊂ B) if every element of A is in B. The empty set is a subset of every set. A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets. The power set P(A) is the set of all subsets of A, so it has 2ⁿ elements.
Be careful to separate an element from a subset: for A = {1, {2}}, the object {2} is an element of A, but {1} is a subset of A.

Operations on Sets

Union A ∪ B: elements in A or B or both. Intersection A ∩ B: common elements. Difference A − B = A ∩ B′. Complement A′ = U − A, where U is the universal set. A and B are disjoint if A ∩ B = φ.
Laws: A ∪ φ = A, A ∩ U = A, De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Counting with Sets

n(A ∪ B) = n(A) + n(B) − n(A ∩ B). n(A − B) = n(A) − n(A ∩ B). For three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C).
For word problems, fill the innermost region of the Venn diagram first, then work outwards.

Solved Examples

Example 1. Write A = {x : x is a prime number less than 12} in roster form.
Answer:
A = {2, 3, 5, 7, 11}

Example 2. How many subsets and proper subsets does {a, b, c, d} have?
Answer:
2⁴ = 16 subsets.
16 subsets and 15 proper subsets.

Example 3. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, find A ∪ B, A ∩ B and A − B.
Answer:
A ∪ B = {1, 2, 3, 4, 5, 6}, A ∩ B = {3, 4}, A − B = {1, 2}.

Example 4. If U = {1, 2, ..., 10} and A = {2, 4, 6, 8, 10}, find A′.
Answer:
A′ = {1, 3, 5, 7, 9}

Example 5. In a class of 40, 25 like cricket and 20 like football, and 8 like both. How many like neither?
Answer:
At least one = 25 + 20 − 8 = 37.
40 − 37 = 3 students

Example 6. If n(A) = 30, n(B) = 25 and n(A ∪ B) = 45, find n(A ∩ B).
Answer:
n(A ∩ B) = 30 + 25 − 45.
10

Example 7. Verify (A ∪ B)′ = A′ ∩ B′ for U = {1, 2, 3, 4}, A = {1, 2}, B = {2, 3}.
Answer:
A ∪ B = {1, 2, 3}, so (A ∪ B)′ = {4}. A′ = {3, 4}, B′ = {1, 4} and A′ ∩ B′ = {4}.
Both sides equal {4}.

Example 8. Find the number of elements in P(P(A)) if A has 2 elements.
Answer:
P(A) has 4 elements and P(P(A)) has 2⁴ elements.
16

Example 9. In a group of 100 persons, 72 speak English and 43 speak French. How many speak English only and how many speak French only (everyone speaks at least one)?
Answer:
Both = 72 + 43 − 100 = 15.
English only = 57, French only = 28.

Example 10. If A ⊂ B, find A ∩ B and A ∪ B.
Answer:
Every element of A is already in B.
A ∩ B = A and A ∪ B = B.