SETS - Unit 17 - Solved Exercise 17.1 - Mathematics For Class X Science

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Unit 17: SETS AND FUNCTIONS
Solved Exercise 17.1

Q.1: Write the following sets in Tabular form.

1. A = Set of all integers between -3 and 3.
2. B = Set of composite numbers less than 11.
3. C = {x | x ∈ P ⋀ 5 < x ≤ 13}
4. D = {y | y ∈ O ⋀ 7 < y < 17}
5. E = {z | z ∈ R ⋀ z² = 121}
6. F = {p | p ∈ Q ⋀ p² = -1}

Ans: SOLUTION:
(i) A = Set of all integers between -3 and 3.
Solutions:
Tabular form A = {-2, -1, 0, 1, 2} Answer

(ii) B = Set of composite numbers less than 11.
Solutions:
Tabular form B = {4, 6, 8, 9, 10} Answer

(iii) C = {x | x ∈ P ⋀ 5 < x ≤ 13}
Solutions:
Tabular form C = {7, 11, 13} Answer

(iv) D = {y | y ∈ O ⋀ 7 < y < 17}
Solutions:
Tabular form D = {9, 11, 13, 15} Answer

(v) E = {z | z ∈ R ⋀ z² = 121}
Solutions:
Tabular form E = {-11, 11} Answer

(vi) F = {p | p ∈ Q ⋀ p² = -1}
Solutions:
Tabular form F = { } or Ø Answer

Q.2: Write the following sets in Builder form.

1. A = Set of all rational numbers between 5 and 6.
2. B = {1, 2, 3, 4, 6, 12}.
3. C = {0, 土1, 土2, .......土40}
4. D = {-4, -2, 0, 2, 4}
5. E = {1, 4, 9, 16, 25}
6. F = {-1, -3, -5, -7, ........}

Ans: SOLUTION:
(i) A = Set of all rational numbers between 5 and 6.
Solution:
Set builder form A = {x | x ∈ Q ∧ 5 < x < 6} Answer

(ii) B = {1, 2, 3, 4, 6, 12}.
Solution:
Set builder from B = {x | x ∈ Z ∧ x is positive divisor of 12} Answer

(iii) C = {0, 土1, 土2, .......土40}
Solution:
Set builder from C = {x | x ∈ Z ∧ -40 ≤ x ≤ 40} Answer

(iv) D = {-4, -2, 0, 2, 4}
Solution:
Set builder from D = {x | x ∈ Z ∧ -4 ≤ x ≤ 4} Answer

(v) E = {1, 4, 9, 16, 25}
Solution:
Set builder from E = {x | x is the square of first five natural numbers}
OR E = {x | x ∈ N ∧ N² ≤ 5} Answer

F = {-1, -3, -5, -7, ........}
Solution:
Set builder from F = {x | x ∈ O ∧ x ≤ -1}
OR F = {x | x is negative odd integers} Answer

3. Write any five examples of empty set.
Ans: EMPTY SETS:
A = {x | x is a letter before 'a' in the English alphabet}.
B = {x | x is less than 7 and greater than 8}
C = {x | x ∈ N ∧ x < 1}
D = {x | x ∈ O ∧ 5 < x < 7}
E = {x | x ∈ R ∧ x² = -1}
F = Set of triangles with four sides.
G = Set of odd numbers divisible by 2

4. Classify the following as finite and infinite sets.

1. Set of Asian countries.
2. Set of all medical universities in the world
3. Set of all real numbers between 6 and 9.
4. Set of all the even prime numbers.
5. Set of all odd numbers less than 5.

SOLUTION:
(i) Set of Asian countries.
Ans: Finite set.

(ii) Set of all medical universities in the world.
Ans: Finite set.

(iii) Set of all real numbers between 6 and 9.
Ans: Infinite set.

(iv) Set of all the even prime numbers.
Ans: Finite set.

(v) Set of all odd numbers less than 5.
Ans: Infinite set.

5. Write an equivalent set, an improper subset and three proper subsets of each of the following sets.

1. P = {a, e, i, o, u}
2. Q = {x | x ∈ Z ∧ -2 ≤ x ≤ 2}

(i) P = {a, e, i, o, u}
Ans: SOLUTION:
Equivalent set = {a, b, c, d, e}
OR {1, 2, 3, 4, 5}
Improper subset = {a, e, i, o, u} or Set of Vowels.
Proper subset = { }, {a}, {e}, {i}, {o}, {u}, {a,e}, {a, i}, {a, o}, {a, u}, {e, i}, {e, o}, {e, u}, {i, o}, {i, u}, {o, u}, {a, e, i}, {a, e, o}, {a, e, u}, (a, i, o}, {a, i, u}, {a, o, u}, {e, i, o}, {e, i, u}, {e, o, u}, {i, o, u}, {a, e, i, o}, {a, e, i, u}, {a, e, o, u}, {a, i, o, u}, {e, i, o, u}
(Note: Above are all the proper subset of set P, select any three)

(ii) Q = {x | x ∈ Z ∧ -2 ≤ x ≤ 2}
Ans: Solution:
Equivalent set = {5, 6, 7, 8, 9}
Improper subset = {-2, -1, 0, 1, 2} 
Proper subset = ∅ or { }, {-2}, {-1}, {0}, {1},  {2}, {-2, -1},  {-2, 0}, {-2, 1}, {-2, 2}, {-1, 0}, {-1, 1}, {-1, 2}, {0, 1}, {0, 2}, {1, 2}, {-2, -1, 0}, {-2,  -1, 1}, {-2, -1, 2}, {-2, 0, 1}, {-2, 0, 2}, {-2, 1, 2}, {-1, 0, 1}, {-1, 0, 2}, {-1, 1, 2}, {0, 1, 2}, {-2, -1, 0, 1}, {-2, -1, 1, 2}, {-2, -1, 0, 2}, {-2, 0, 1, 2}, {-1, 0, 1, 2}
(Note: Above are all the proper subset of set P, select any three)

6. Write any two examples of singleton in set builder form.
Ans: Singleton Set:
Example 1: {x | x ∈ N Λ x² = 25} Answer
Example 2: {x | x ∈ Z Λ -1 < x < 1} Answer

7. Write power sets of the following sets.

1. A = {5, 10, 15}
2. B = {x | x ∈ Z Λ -1 < x < 4}

(i) A = {5, 10, 15}
Solution:
(For Power set, 'n' is the number of elements of set A.)
n|P(A)| = 2ⁿ = 2³ = 2 x 2 x 2 = 8
∴ [P (A)] = {∅ or { }, {5}, {10}, {15}, {5, 10}, {5, 15}, {10, 15}, {5, 10, 15}} Ans.

(ii) B = x | x ∈ Z Λ -1 < x < 4}
Solution:
(For Power set, 'n' is the number of elements of set B.)
B = {0, 1, 2, 3}
n|P(B)| = 2ⁿ = 2⁴ = 2 x 2 x 2 x 2 = 16
∴ [P (B)] = {∅ or { }, {0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, { 1, 3}, { 2, 3}, {0, 1, 2 }, {0, 1, 3}, {0, 2, 3},  {1. 2, 3}, {0, 1, 2, 3}}Ans.

8. Find a set which has only

1. Two proper subset
2. One proper subset
3. no proper subset

(i) Two proper subset
Answer: There is no such set which has two proper subset only.

(ii) One proper subset
Answer: (Note: Any set contain one element has one proper subset because it has null set also)
1. {x | x ∈ Z Λ -1 < x < 1} OR {0}
2. {a}

(iii) no proper subset
Answer: ∅ or { } has no proper subset

EXAMPLES FROM TEXT BOOK