Question 1: Which of the following collections of objects are sets?

1. All the months in a year
• Answer: YES. It is a well-defined collection of distinct objects. There are 12 months in a year. So it is a definite set of elements.
• A = {January, February, March, April, May, June, July, August, September, October, November, December}
2. All the rivers flowing in UP
• Answer: YES. It is a well-defined collection of distinct objects.
• A = {Ganga, Yamuna…}
3. All the planets in our solar system
• Answer: YES. It is a well-defined collection of distinct objects.
• A= {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
4. All the interesting dramas written by Premchand
• Answer: NO. It is not a well-defined collection of objects.  Had it been “All dramas written by Premchand” then it could have been a set as that would be well defined.
5. All the short boys in your class
• Answer: NO. It is not a well-defined collection of objects.  Had it been “All boys shorter than 5’3”, then it would have become well defined. We need a benchmark to have a well-defined list of objects.
6. All the letters of the English Alphabets which precedes K
• Answer: YES. It is a well-defined collection of distinct objects.
• A = {A, B, C, D, E, F, G, H, I, J}
7. All the pet dogs in Nagpur
• Answer: YES. It is a well-defined collection of distinct objects. Technically, all pets in Nagpur should be registered with a Government department. This might be a big list but still a well-defined list of objects.
8. All the dishonest shop owners in Noida
• Answer: No. Not well defined.
9. All the students in your school with age exceeding 15 years
• Answer: YES. It is a well-defined collection of distinct objects. School has a list of students that study there. The set would include all whose age is more than 15 years.
10. All the girls of Gita’s class who are taller than Gita
• Answer: YES. It is a well-defined collection of distinct objects. We know all the girls studying in the class, we know Gita’s height and hence the set would contain distinct girls taller than Gita.

Question 2: Rewrite the following statements using the set notations:

1. p is an element of A
2. q does not belong to set B
3. a and b are members of set C
• Answer: a, b ∈ C
4. B and C are equivalent sets
5. Cardinal number of set E is 15
• Note: The number of distinct elements contained in a finite set A is called the cardinal number of A and is denoted by n(A).
• For example if A = {1, 2, 3, 4, 5}, then n(A)=5
6. A is an empty set and B is a non-empty set
• Answer: A = φ and B ≠ φ
• Note: A set consisting of no elements is called an empty set or a null set or a void set. It is denoted by φ (called phai). We write f = { }
7. 0 is a whole number, but 0 is not a natural number
• Answer: 0 ∈ W but 0 ∉ N

Note: Whole numbers (W): The numbers {0, 1, 2, 3, …}.

Natural numbers (N): The counting numbers {1, 2, 3, …}, are called natural numbers

Question 3: Describe the following sets in roster form

1. B = {x | x ∈ W, x <= 6}
• Answer: B = {0, 1, 2, 3, 4, 5, 6}
2. C = {x | x is a factor of 32}
• Answer: C = {1, 2, 4, 8, 16, 32}
• Note: You can calculate the factors using a tree method.
3. E = {x | x = (2n+1), n ∈ W, n <= 4}
• Answer: E = {1, 3, 5, 7, 9}
• Note: n ∈  W and n is <= 4. This means n is 0, 1, 2, 3, and 4. Now calculate x based on the given formula. Example: When n = 0, x = 1.
4. F = {x | x = n2, n ∈ N, 2 <= n <= 5}
• Answer: F = {4, 9, 16, 25}
• Note: n ∈ N and 2 <= n <= 5. This means n is 2,3,4,5. Now substitute the value of n to calculate x.
5. G = {x | x = {n/(n+3)}  , n ∈ N and n <=5}
• Answer: G = {1/4, 2/5, 3/6. 4/7, 5/8}
• Note: n ∈ N and n <=5, which means n is 1, 2, 3, 4, 5. Now substitute the value of n in the formula given for x.
6.  H = {x | x is a two digit number, the sum of the digits is 8}
• Answer: H = {17, 26, 35, 44, 53, 62, 71, 80}
• Note: If the first digit is x, then the second digit is (8-x). So now try the values of x as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10…. x cannot be 0 as it is a two digit number. Also by same logic, x cannot be greater than 8 either. Hence x can only be 1, 2, 3, 4, 5, 6, 7, and 8 only.
7. I = {x | x ∈ N, x is divisible by both 4 and 6 and x <= 60}
• Answer: I = {12, 24, 36, 48, 60}
• Note: x <= 60 means that x = {1, 2, 3…60}. Of these numbers x needs to be divisible both by 4 and 6.
8. J = { x | x = (1/n) , n ∈ N and n <=5}
• Answer: J = {1,  , , , }
• Note: n ∈ N and n <=5 which means that n is 1, 2, 3, 4, 5. Now substitute the values.
9. L = {x | x is a letter of the word ‘careless’}
• Answer: L = {c, a, r, e, l, s}
• Note: In careless, the alphabet e and s are repeated. We only need to take into account distinct elements only.

Question 4: Describe the following sets in set builder form

1. A = { 5, 6, 7, 8, 9, 10, 11, 12}
• Answer:  A = {x | x ∈ N, 4 < x < 13} or A = {x | x ∈ N, 5 <= x <=12}
2. B = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}
• Answer:  B = {x | x is a factor of 48}
3. C = {11, 13, 17, 19, 23, 29, 31, 37}
• Answer:  C = {x | x is a prime number, x ∈ N and 10 < x < 30}
4. D = {21, 23, 25, 27, 29, 31, 33, 35, 37}
• Answer:  D = {x | x is an odd number, 20 < x < 38} or
• D = {x | x = (2n+19), n ∈ N, and 1 <= n <=9}
5. I = {9, 16, 25, 36, 49, 64, 81, 100}
• Answer: I = {x | x=n2, n ∈ N, and 3 <= n <=10}
• Note: If you notice, the numbers are square of 3, 4, 5, 6, 7, 8, 9, 10.
6. J = { -2, 2}
• Answer:  J = {x | x ∈ Z, x2 = 4}
• Note: Integers (Z): Positive and negative counting numbers, as well as zero:{…, -2, -1, 0, 1, 2,…}.
• x2=4 which is x =4 which means x can be 2 or -2
7. K = {0}
• Answer:  K = {x | x = 0}
8. L = { }
• Answer: L = {x | x ∈ N, x ≠ x}
9. M = {a, b, c, d, e, f, g, h, i}
• Answer: M = {x | x is an English alphabet which precedes J}
10. P = {2/7, 3/8,  4/9,  5/10,  6/11,  7/12,  8/13,  9/14}
• Answer:  P = {x | x = , n ∈ N, 2 <= n <= 9}
11. S = {Atlantic, Artic}
• Answer:  S = {x | x is an ocean name that starts with A}
12. T = {Mars, Mercury}
• Answer:  T = {x | x is a planet whose name starts with an M}

Question 5: Separate finite and infinite sets from the following:

1. Set of leaves on a tree
• Answer: Finite. This is because in this case, the process of counting the leaves would surely come to an end.
2. Set of all counting numbers
• Answer: Infinite. This is because there is no end to the numbers.
3. {x | x ∈ N, x > 1000}
• Answer: Infinite. This is because there is no end to the numbers since x > 1000.
4. {x | x ∈ W, x < 5000}
• Answer: Finite. W is whole numbers which are all natural numbers including 0. Since x is less than 5000, there are finite numbers to be counted.
5. {x | x ∈ Z, x < 4 }
• Answer: Infinite. Z is an integer. Integers can be negative numbers too. So in this case, though x is limited to less than 4 on the positive scale, it can go to infinity on the negative scale.
6. Set of all triangles in a plane
• Answer: Infinite. This is because there can be uncountable number of triangles in a plane.
7. Set of all points on a circumference of a circle
• Answer: Infinite. This is because there can be uncountable number of points on a circumference of a circle.
8. {1, 2, 3, 1, 2, 3, 1, 2, 3, …}
• Answer: Infinite. This is because the set is not ending. It will keep having 1, 2, 3 repeated again and again
9. {x | x ∈ Q, 2 < x< 3}
• Note: Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer.[1] All integers are rational, but the converse is not true.

Question 6: Which of the following are empty sets?

1. A = { x | x ∈ N, x + 5 = 5}
• Note: Natural Number N = {1, 2, 3, 4, 5…}. Since x+5 = 5 is given, it means, x = 0. But 0 does not belong to N and hence the set it empty or null.
2. B = { x | x ∈ N, 2x + 3 = 6}
• Note: Natural Number N = {1, 2, 3, 4, 5…}. Since 2x+3 = 6 is given, it means, x = 1.5. But 1.5 does not belong to N and hence the set it empty or null.
3. C = { x | x ∈ W, x + 2 < 5}
• Answer: Not an Empty Set
• Note: Whole Number W = {0, 1, 2, 3, 4, 5…}. Since x+2 < 5 is given, it means, x < 3. Therefore x can be 0, 1, 2 and hence it is not an Empty Set.
4. D = { x | x ∈ N, 1 < x <= 2}
• Answer: Not an Empty Set
• Note: Natural Number N = {1, 2, 3, 4, 5…}. Since 1 < x <=2, it means that x can be 2. Hence D is not an Empty Set.
5. E = { x | x ∈ N, x2 + 4 = 0}
• Note: Natural Number N = {1, 2, 3, 4, 5…}.  x2 + 4 = 0 means that x2 is – 4 which is not possible.
6. F = { x | x is a prime number, 90 < x< 96}
• Note: If 90 < x< 96, this means x can be 91, 92, 93, 94, 95. All these numbers are divisible by a number. E.g. 91 is divisible by 7, 92 by 2, 93 by 3 and so on.
7. G = {x | x is an even prime}
• Answer: Not an Empty Set.
• Note: 2 is an even prime number.

Question 7: Which of the following are pairs of equivalent sets?

Note: Two finite sets A and B are said to be equivalent, if n(A) = n(B), that is they have the same number of elements. An equivalent set is simply a set with an equal number of elements. The sets do not have to have the same exact elements, just the same number of elements.

1. A = {2, 3, 5, 7} and B = {x : x is a whole number, x < 3}
• Note: A = {2, 3, 5, 7} and B = {0, 1, 2} n(A) = 4 while n(B) = 3
2. C = {x : x + 2 = 2} and D =φ
• Note: C = {0} and D = {} Therefore n(C) = 1 while n(D) = null
3. E = {x : x is a natural number, x < 4} and F = {x : x is a whole number, x <  3}
• Note: E = { 1, 2, 3} and F = { 0, 1, 2} Therefore n(E) = 3 while n(F) = 3 Hence E and F are equivalent sets.
4. G = {x : x is an integer, – 3 < x < 3} and H = {x : x is a factor of 16}
• Note: G = {-2, -1, 0, 1, 2} and H = {1, 2, 4, 8, 16} Therefore n(G) = 5 and n(H) = 5 Hence they are equivalent sets

Question 8: State whether the following statements are true or false

1. {a, b, c, 1, 2, 3} is not a set
• Answer: False. It is a set. It contains elements.
2. {5, 7, 9} = {9, 5, 7}
• Answer: True. Both sets contain the same elements just in different order. The order of elements does not matter.
3. {x | x ∈ W, x + 8 = 8} is a singleton set
• Answer: True. The only element in the set would be 0. Hence it is a singleton set.
• Note: a singleton, also known as a unit set, is a set with exactly one element. For example, the set {0} is a singleton. The term is also used for a 1-tuple (a sequence with one element).
4. {x | x ∈ W, x < 0} = f
• Answer: True. The set is a null set as there are no elements.
5. {x | x ∈ N, x + 5 = 3} = f
• Answer: True. x = -2 but x is a natural number. Hence a null set.
6. {x | x ∈ N, 3 < x <= 4} = f
• Answer: False.  There is a valid element in the set which is 4.
7. If A = {x | x is a letter of the work, ‘Meerut’} then n(A)=6