In symbol, we write x ⊆ y
Reading Notation:
Read ⊆ as "X is a subset of Y" or "X is contained in Y"
Read ⊈ as "X is a not subset of Y" or "X is not contained in Y".
The formula for calculating the number of subsets = 2^{n}, where n = number of elements in the set.
Note:
 If A and B are any two nonempty sets such that A ⊆ B Let x be any element such that x ∈ A ⇒ x ∈ B
 Every set is a subset of itself that means A ⊆ A
 An empty set i.e ∅ is a subset of every set
Intervals as subsets of R:
Example.1 Which of the following are correct statements?
A = {1, 2, 3, 4, 5, 6}
(a) {2, 3} ⊂ A
(b) {1, 2, 3, 4, 5,6, 7} ⊃ A
(c) 8 ⊂ A
(d) {3, 5, 1, 7} ⊃ A
(e) {1} ⊂ A
(f) {1, 2, 3, 4} ⊂ A
(g) { } ⊃ A
(h) ϕ ⊂ A
Ans. (a), (b), (e), (f), (h)
Subsets are classified as:
A proper subset is one that contains a few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.
Example: If set A = {2, 4, 6}, then,
Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.
Proper Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
Improper Subset: {2,4,6}
1. Proper Subsets
How many subsets and proper subsets does a set have?
2. Improper Subset
Note: The empty set is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other set.
Q.1. How many elements are there in power set if
(a) A = {ϕ}
(b) B = {a, b}
(c) C = {l, m, n}
(d) D = {4, 9}
Ans.
(a) Here n(A) = 0 ,so n(P(A)) = 2^{0} = 1
(b) 4
(c) 8
(d) 4
Q.2. Find the number of proper subsets of the following.
(a) P = {x : x ∈ N, x < 5}
(b) Q = {x : x is an even prime number}
(c) R = {x : x ∈ W, x < 2}
(d) T = { }
(e) X = {0}
(f) Y = {x : x is prime, 2 < x < 10}
Ans.
(a) 15
(b) 1
(c) 3
(d) 0
(e) 1
(f) 7
Q.3. Write down all the subsets of
(a) {8}
(b) {p, q}
(c) {1, 3, 5}
(d) ϕ
Ans.
(a) ∅, {8}
(b) ∅, {p}, {q}, {p, q}
(c) ∅, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}
(d) ∅
Q.4. What universal sets would you propose for the following?
(a) The set of all squares
(b) The set of all even numbers
(c) The set of all isosceles triangles
(d) The set of all negative integers
(e) The set of all prime numbers
(f) The set of all obtuse angled triangles
Ans.
(a) {quadrilaterals}
(b) N
(c) {triangles}
(d) I
(e) N
(f) {triangles}
Q.5. Fill in the blank spaces using the symbols ⊂ or ⊄.
(a) {1, 2, 3} ______ {1, 3, 5}
(b) ϕ ______ {4, 7, 9}
(c) {x : x is rectangle in a plane} ______ {x : x is a quadrilateral in a plane}
(d) {x : x is an odd natural number} ______{x : x is an integer}
(e) {x : x is a prime number} ______ {x: x is a composite number}
(f) {5, 10, 15, 20, 25, 30} ______ {10, 20, 30, 40}
Ans.
(a) ⊄
(b) ⊂
(c) ⊂
(d) ⊂
(e) ⊄
(f) ⊄
Q.6. Given set A = {a, b, c) B = {p, q, r} C = {x, y, z, m, n, t} which of following are considered as universal set for all the three sets.
(a) P = {a, b, c, p, q, x, y, m, t}
(b) Q = {ϕ}
(c) R = {a, c, q, r, b, p, t, z, m, n, x, y}
(d) S = {b, c, q, r, n, t, p, q, x, m, y, z, f, g}
Ans. (c) as it contains all the elements of sets A, B & C.
Q.7. Let A be the set of letters of the word FOLLOW. Find:
(a) A
(b) n(A)
(c) Number of subsets of A
(d) Number of proper subsets of A
(e) Power set of A
Ans.
(a) {F, O, L, W}
(b) 4
(c) 16
(d) 15
(e) {∅, {F}, {O}, {L}, {W}, {F, O}, {F, L}, {F, W}, {O, L}, {O, W}, {L, W}, {F, O, L}, {F, O, W}, {F, L, W}, {O, L, W}, {F, O, L, W}}
Q.8. Find the power set of the following sets.
(a) A = {a, b, c}
(b) B = {0, 7}
(c) C = {0, 5, 10}
(d) D = {x}
Ans.
(a) {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}
(b) {∅, {0}, {7}, {0, 7}}
(c) {∅, {0}, {5}, {10}, {0, 5}, {0, 10}, {5, 10}, {0, 5, 10}}
(d) {∅, {x}}
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