State whether the given set is infinite or finite:
(i) { 3, 5, 7, ...... }
(ii) { 1, 2, 3, 4 }
(iii) { ..., -3, -2, -1, 0, 1, 2 }
(iv) { 20, 30, 40, 50, ..., 200 }
Answer
(i) The dots after 7 show that the elements continue without end.
Infinite set
(ii) The set has a definite (countable) number of elements, i.e. 4 elements.
Finite set
(iii) The dots before -3 show that the elements continue without end on the left.
Infinite set
(iv) The elements start at 20 and end at 200, so the number of elements is countable.
Finite set
Which of the following sets is empty?
(i) Set of counting numbers between 5 and 6.
(ii) Set of odd numbers between 7 and 19.
(iii) Set of odd numbers between 7 and 9.
(iv) Set of even numbers that are not divisible by 2.
(v) { 0 }
Answer
(i) There is no counting number between 5 and 6. So, this set is empty.
(ii) The odd numbers between 7 and 19 are 9, 11, 13, 15 and 17. So, this set is not empty.
(iii) There is no odd number between 7 and 9 (the only number between them is 8, which is even). So, this set is empty.
(iv) Every even number is divisible by 2, so there is no even number that is not divisible by 2. So, this set is empty.
(v) The set { 0 } has one element, namely 0. So, this set is not empty.
Hence, the sets in (i), (iii) and (iv) are empty.
State which pair of sets given below are equal sets and which are equivalent:
(i) { 3, 5, 7 } and { 5, 3, 7 }
(ii) { 8, 6, 10, 12 } and { 3, 2, 4, 6 }
(iii) { 7, 7, 2, 1, 2 } and { 1, 2, 7 }
(iv) { 2, 4, 6, 8, 10 } and { a, b, d, e, m }
Answer
(i) Both sets have exactly the same elements 3, 5 and 7.
Equal sets
(ii) Both sets have 4 elements each, but the elements are not the same.
Equivalent sets
(iii) { 7, 7, 2, 1, 2 } = { 1, 2, 7 } (repetition does not change a set), which has the same elements as { 1, 2, 7 }.
Equal sets
(iv) Both sets have 5 elements each, but the elements are not the same.
Equivalent sets
State which of the following are finite sets and which are infinite:
(i) Set of integers
(ii) { Multiples of 5 }
(iii) { Fractions between 1 and 2 }
(iv) { Number of people in India }
(v) Set of trees in the world
(vi) Set of leaves on a tree
(vii) Set of children in all the schools of Delhi
(viii) { ...., -4, -2, 0, 2, 4, 6, 8 }
(ix) { -12, -9, -6, -3, 0, 3, 6, ..... }
(x) { Number of points in a line segment 4 cm long }
Answer
(i) The integers are ......, -2, -1, 0, 1, 2, ......, which continue without end.
Infinite set
(ii) The multiples of 5 are 5, 10, 15, 20, ......, which continue without end.
Infinite set
(iii) Between any two numbers there are unlimited fractions.
Infinite set
(iv) The number of people in India is large but countable.
Finite set
(v) The number of trees in the world cannot be counted exactly.
Infinite set
(vi) The number of leaves on a tree is countable.
Finite set
(vii) The number of children in all the schools of Delhi is countable.
Finite set
(viii) The dots before -4 show that the elements continue without end on the left.
Infinite set
(ix) The dots after 6 show that the elements continue without end on the right.
Infinite set
(x) A line segment, however small, contains an unlimited number of points.
Infinite set
State whether or not the following sets are empty:
(i) { Prime numbers divisible by 2 }
(ii) { Negative natural numbers }
(iii) { Women with height 5 metres }
(iv) { Integers less than 5 }
(v) { Prime numbers between 17 and 23 }
(vi) Set of even numbers not divisible by 2
(vii) Set of multiples of 3 that are more than 9 and less than 15.
Answer
(i) 2 is a prime number which is divisible by 2. So, the set = { 2 }.
Not empty
(ii) The natural numbers are 1, 2, 3, ......; none of them is negative. So, there is no negative natural number.
Empty
(iii) There is no woman with a height of 5 metres.
Empty
(iv) The integers less than 5 are 4, 3, 2, 1, 0, -1, ......, which exist.
Not empty
(v) 19 is a prime number between 17 and 23. So, the set = { 19 }.
Not empty
(vi) Every even number is divisible by 2, so there is no even number that is not divisible by 2.
Empty
(vii) The only multiple of 3 that is more than 9 and less than 15 is 12. So, the set = { 12 }.
Not empty
State if the given pairs of sets are equal sets or equivalent sets:
(i) { Natural numbers less than five } and { Letters of the word 'BOAT' }.
(ii) { 2, 4, 6, 8, 10 } and { even natural numbers less than 12 }.
(iii) { 1, 3, 5, 7, ..... } and set of odd natural numbers.
(iv) { Letters of the word MEMBER } and { Letters of the word 'REMEMBER' }.
(v) { Negative natural numbers } and { 50th day of a month }
(vi) { Even natural numbers } and { Odd natural numbers }.
Answer
(i) { Natural numbers less than five } = { 1, 2, 3, 4 } and { Letters of the word 'BOAT' } = { B, O, A, T }. Both have 4 elements each, but the elements are not the same.
Equivalent sets
(ii) { 2, 4, 6, 8, 10 } and { even natural numbers less than 12 } = { 2, 4, 6, 8, 10 } have exactly the same elements.
Equal sets
(iii) { 1, 3, 5, 7, ..... } and the set of odd natural numbers = { 1, 3, 5, 7, ..... } have exactly the same elements.
Equal sets
(iv) { Letters of the word MEMBER } = { M, E, B, R } and { Letters of the word 'REMEMBER' } = { R, E, M, B } have exactly the same elements.
Equal sets
(v) { Negative natural numbers } = { } (there is no negative natural number) and { 50th day of a month } = { } (no month has a 50th day). Both are empty sets, and two empty sets are always equal.
Equal sets
(vi) { Even natural numbers } and { Odd natural numbers } are both infinite sets, so they have the same (infinite) number of elements, but the elements are not the same.
Equivalent sets
State whether the following are finite or infinite sets:
(i) { 2, 4, 6, 8, ...., 800 }
(ii) { ...., -5, -4, -3, -2 }
(iii) { x : x is an integer between -60 and 60 }
(iv) { No. of electrical appliances working in your house }
(v) { x : x is a whole number greater than 20 }
(vi) { x : x is a whole number less than 20 }
Answer
(i) The elements start at 2 and end at 800, so the number of elements is countable.
Finite set
(ii) The dots before -5 show that the elements continue without end on the left.
Infinite set
(iii) The integers between -60 and 60 are -59, -58, ......, 58, 59, which are countable.
Finite set
(iv) The number of electrical appliances working in your house is countable.
Finite set
(v) The whole numbers greater than 20 are 21, 22, 23, ......, which continue without end.
Infinite set
(vi) The whole numbers less than 20 are 0, 1, 2, ......, 19, which are countable.
Finite set
For each statement given below, write True or False:
(i) { ...., -8, -4, 0, 4, 8 } is a finite set.
(ii) { -32, -28, -24, -20, ...., 0, 4, 8, 16 } is an infinite set.
(iii) { x : x is a natural number less than 1 } is the empty set.
(iv) { Whole numbers between 15 and 16 } = { Natural numbers between 5 and 6 }.
(v) { Odd numbers divisible by 2 } is the empty set.
(vi) { Even natural numbers divisible by 3 } is the empty set.
(vii) { x : x is positive and x < 0 } is the empty set.
(viii) { ...., -5, -3, -1, 1, 3, 5, .... } is a finite set.
Answer
(i) The dots before -8 show that the elements continue without end, so the set is infinite, not finite.
False
(ii) The elements start at -32 and end at 16, so the number of elements is countable. So, it is a finite set, not an infinite set.
False
(iii) There is no natural number less than 1, so the set has no element.
True
(iv) There is no whole number between 15 and 16, and there is no natural number between 5 and 6. So, both sets are the empty set and are therefore equal.
True
(v) No odd number is divisible by 2, so the set has no element.
True
(vi) 6, 12, 18, 24, ...... are even natural numbers divisible by 3, so the set is not empty.
False
(vii) No positive number can be less than 0, so the set has no element.
True
(viii) The dots on both sides show that the elements continue without end, so the set is infinite, not finite.
False
State, giving reasons, which of the following pairs of sets are disjoint sets and which are overlapping sets:
(i) A = { Girls with ages below 15 years } and
B = { Girls with ages above 15 years }
(ii) C = { Boys with ages above 20 years } and
D = { Boys with ages above 27 years }
(iii) A = { Natural numbers between 35 and 60 } and
B = { Natural numbers between 50 and 80 }
(iv) P = { Students of Class IX studying in I.C.S.E. Board } and
Q = { Students of Class IX }
(v) A = { Natural numbers that are multiples of 3 and less than 30 } and
B = { Natural numbers divisible by 4 and lying between 20 and 45 }
(vi) P = { Letters in the word 'ALLAHABAD' } and
Q = { Letters in the word 'NAINITAL' }
Answer
Overlapping sets : Two sets are overlapping if they have at least one element in common.
Disjoint sets : Two sets are disjoint if they have no element in common.
(i) Disjoint sets; as no girl can be of age below 15 years and also above 15 years, the two sets have no element in common.
(ii) Overlapping sets; as boys with ages above 27 years are also above 20 years, so they are common to both the sets.
(iii) Overlapping sets; as the natural numbers from 51 to 59 are common to both the sets.
(iv) Overlapping sets; as the students of Class IX studying in I.C.S.E. Board are common to both the sets.
(v) Overlapping sets; as A = { 3, 6, 9, ......, 27 } and B = { 24, 28, 32, 36, 40, 44 }. The number 24 is common to both the sets.
(vi) Overlapping sets; as P = { A, L, H, B, D } and Q = { N, A, I, T, L }. The letters A and L are common to both the sets.