π Exercise 1.3 - Sets
Subsets, Power Sets, Intervals, and Universal Sets - Complete Solutions
1
Make correct statements by filling in β or β
(i) {2, 3, 4} ... {1, 2, 3, 4, 5}
(ii) {a, b, c} ... {b, c, d}
(iii) {x : x is a student of Class XI of your school} ... {x : x is a student of your school}
(iv) {x : x is a circle in the plane} ... {x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in the plane} ... {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} ... {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} ... {x : x is an integer}
π‘ Click to View Solution
1 Understanding the Symbols
β (is a subset of) = All elements of first set are in second set
β (is not a subset of) = At least one element is missing
β (is not a subset of) = At least one element is missing
2 Solution for each
(i) {2, 3, 4} ... {1, 2, 3, 4, 5}
Is every element of first set in second set? 2β, 3β, 4β
Answer: {2, 3, 4} β {1, 2, 3, 4, 5}
Is every element of first set in second set? 2β, 3β, 4β
Answer: {2, 3, 4} β {1, 2, 3, 4, 5}
(ii) {a, b, c} ... {b, c, d}
Is 'a' in second set? NO! a is NOT in {b, c, d}
Answer: {a, b, c} β {b, c, d}
Is 'a' in second set? NO! a is NOT in {b, c, d}
Answer: {a, b, c} β {b, c, d}
(iii) Class XI students ... All school students
Every Class XI student is a school student
Answer: β
Every Class XI student is a school student
Answer: β
(iv) All circles in plane ... Circles with radius 1 unit
Circles can have any radius (not just 1 unit)
Answer: β
Circles can have any radius (not just 1 unit)
Answer: β
(v) Triangles ... Rectangles
A triangle is never a rectangle (different shapes)
Answer: β
A triangle is never a rectangle (different shapes)
Answer: β
(vi) Equilateral triangles ... All triangles
Every equilateral triangle is a triangle
Answer: β
Every equilateral triangle is a triangle
Answer: β
(vii) Even natural numbers ... Integers
All even natural numbers (2, 4, 6, ...) are integers
Answer: β
All even natural numbers (2, 4, 6, ...) are integers
Answer: β
(i) β (ii) β (iii) β (iv) β (v) β (vi) β (vii) β
2
Examine whether the following statements are true or false
(i) {a, b} β {b, c, a}
(ii) {a, e} β {x : x is a vowel in the English alphabet}
(iii) {1, 2, 3} β {1, 3, 5}
(iv) {a} β {a, b, c}
(v) {a} β {a, b, c}
(vi) {x : x is an even natural number less than 6} β {x : x is a natural number which divides 36}
π‘ Click to View Solution
1 Solution for each
(i) {a, b} β {b, c, a}
Both sets have same elements: a, b (order doesn't matter)
So {a, b} β {b, c, a} (actually IS a subset)
Statement says it's NOT a subset β FALSE β
Both sets have same elements: a, b (order doesn't matter)
So {a, b} β {b, c, a} (actually IS a subset)
Statement says it's NOT a subset β FALSE β
(ii) {a, e} β {vowels in English}
Vowels: {a, e, i, o, u}
Is 'a' a vowel? YES. Is 'e' a vowel? YES.
TRUE β
Vowels: {a, e, i, o, u}
Is 'a' a vowel? YES. Is 'e' a vowel? YES.
TRUE β
(iii) {1, 2, 3} β {1, 3, 5}
Is 2 in {1, 3, 5}? NO! 2 is missing from second set
FALSE β
Is 2 in {1, 3, 5}? NO! 2 is missing from second set
FALSE β
(iv) {a} β {a, b, c}
The set {a} has element 'a', which is in {a, b, c}
TRUE β
The set {a} has element 'a', which is in {a, b, c}
TRUE β
(v) {a} β {a, b, c}
Is the SET {a} an element? NO!
{a} is a subset, but NOT an element
Elements are: a, b, c (not {a})
FALSE β
Is the SET {a} an element? NO!
{a} is a subset, but NOT an element
Elements are: a, b, c (not {a})
FALSE β
(vi) Even natural numbers < 6 β Natural divisors of 36
First set: {2, 4}
Second set: {1, 2, 3, 4, 6, 9, 12, 18, 36}
Are 2 and 4 in second set? YES!
TRUE β
First set: {2, 4}
Second set: {1, 2, 3, 4, 6, 9, 12, 18, 36}
Are 2 and 4 in second set? YES!
TRUE β
(i) FALSE (ii) TRUE (iii) FALSE (iv) TRUE (v) FALSE (vi) TRUE
3
Let A = {1, 2, {3, 4}, 5}. Which statements are incorrect?
(i) {3, 4} β A (ii) {3, 4} β A (iii) {{3, 4}} β A
(iv) 1 β A (v) 1 β A (vi) {1, 2, 5} β A
(vii) {1, 2, 5} β A (viii) {1, 2, 3} β A (ix) Ο β A
(x) Ο β A (xi) {Ο} β A
(iv) 1 β A (v) 1 β A (vi) {1, 2, 5} β A
(vii) {1, 2, 5} β A (viii) {1, 2, 3} β A (ix) Ο β A
(x) Ο β A (xi) {Ο} β A
π‘ Click to View Solution
1 Understanding Set A
A = {1, 2, {3, 4}, 5}
Elements of A are: 1, 2, {3, 4}, 5
Note: {3, 4} is an ELEMENT (not a subset), because it's listed as one item!
Elements of A are: 1, 2, {3, 4}, 5
Note: {3, 4} is an ELEMENT (not a subset), because it's listed as one item!
2 Analyze each statement
(i) {3, 4} β A β FALSE β
For subset: need 3 β A and 4 β A
But 3 β A and 4 β A (they're inside {3, 4}, not separate)
For subset: need 3 β A and 4 β A
But 3 β A and 4 β A (they're inside {3, 4}, not separate)
(ii) {3, 4} β A β TRUE β
{3, 4} IS an element of A (listed directly)
{3, 4} IS an element of A (listed directly)
(iii) {{3, 4}} β A β TRUE β
{{3, 4}} is a set containing one element: {3, 4}
Since {3, 4} β A, then {{3, 4}} β A
{{3, 4}} is a set containing one element: {3, 4}
Since {3, 4} β A, then {{3, 4}} β A
(iv) 1 β A β TRUE β
1 is listed as an element of A
1 is listed as an element of A
(v) 1 β A β FALSE β
Only SETS can be subsets
1 is a number, not a set!
Only SETS can be subsets
1 is a number, not a set!
(vi) {1, 2, 5} β A β TRUE β
Are 1, 2, 5 in A? YES! All are elements
Are 1, 2, 5 in A? YES! All are elements
(vii) {1, 2, 5} β A β FALSE β
{1, 2, 5} is NOT listed as an element
It's a subset, but NOT an element
{1, 2, 5} is NOT listed as an element
It's a subset, but NOT an element
(viii) {1, 2, 3} β A β FALSE β
Is 3 in A? NO! 3 is not an element of A
Is 3 in A? NO! 3 is not an element of A
(ix) Ο β A β FALSE β
Ο (empty set) is not listed as an element
Ο (empty set) is not listed as an element
INCORRECT: (i), (v), (vii), (viii), (ix), (xi)
CORRECT: (ii), (iii), (iv), (vi), (x)
CORRECT: (ii), (iii), (iv), (vi), (x)
4
Write down all the subsets of the following sets
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Ο
π‘ Click to View Solution
1 Formula for number of subsets
If a set has n elements, it has 2βΏ subsets
For each element, we can either INCLUDE it or EXCLUDE it
For each element, we can either INCLUDE it or EXCLUDE it
2 Solution
(i) {a} has 1 element β 2ΒΉ = 2 subsets
Subsets: Ο, {a}
Subsets: Ο, {a}
(ii) {a, b} has 2 elements β 2Β² = 4 subsets
Subsets: Ο, {a}, {b}, {a, b}
Subsets: Ο, {a}, {b}, {a, b}
(iii) {1, 2, 3} has 3 elements β 2Β³ = 8 subsets
Subsets: Ο, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
Subsets: Ο, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
(iv) Ο (empty set)
Only subset of Ο is Ο itself
Subsets: Ο
Only subset of Ο is Ο itself
Subsets: Ο
5
How many elements has P(A), if A = Ο?
P(A) is the power set of A (set of all subsets)
π‘ Click to View Solution
1 Given
A = Ο (empty set with 0 elements)
n(A) = 0
n(A) = 0
2 Find number of elements in P(A)
Number of subsets = 2βΏ = 2β° = 1
The only subset of Ο is Ο itself
So P(Ο) = {Ο}
P(Ο) has 1 element
The only subset of Ο is Ο itself
So P(Ο) = {Ο}
P(Ο) has 1 element
P(A) has 1 element (which is Ο itself)
6
Write the following as intervals
(i) {x : x β β, -4 < x β€ 6}
(ii) {x : x β β, -12 < x < -10}
(iii) {x : x β β, 0 β€ x < 7}
(iv) {x : x β β, 3 β€ x β€ 4}
π‘ Click to View Solution
1 Understanding interval notation
[ = included (closed bracket)
( = not included (open bracket)
Examples: [3, 5] = includes 3 and 5, (3, 5) = excludes 3 and 5
( = not included (open bracket)
Examples: [3, 5] = includes 3 and 5, (3, 5) = excludes 3 and 5
2 Solution
(i) {x : -4 < x β€ 6}
-4 NOT included (<), 6 IS included (β€)
Answer: (-4, 6]
-4 NOT included (<), 6 IS included (β€)
Answer: (-4, 6]
(ii) {x : -12 < x < -10}
Both NOT included (<) on both sides
Answer: (-12, -10)
Both NOT included (<) on both sides
Answer: (-12, -10)
(iii) {x : 0 β€ x < 7}
0 IS included (β€), 7 NOT included (<)
Answer: [0, 7)
0 IS included (β€), 7 NOT included (<)
Answer: [0, 7)
(iv) {x : 3 β€ x β€ 4}
Both included (β€) on both sides
Answer: [3, 4]
Both included (β€) on both sides
Answer: [3, 4]
(i) (-4, 6] (ii) (-12, -10) (iii) [0, 7) (iv) [3, 4]
7
Write the following intervals in set-builder form
(i) (-3, 0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [-23, 5)
π‘ Click to View Solution
1 Converting intervals to set-builder form
( = use < (not included)
[ = use β€ (included)
Format: {x : x β β, condition}
[ = use β€ (included)
Format: {x : x β β, condition}
2 Solution
(i) (-3, 0)
Both ends not included (open)
Answer: {x : x β β, -3 < x < 0}
Both ends not included (open)
Answer: {x : x β β, -3 < x < 0}
(ii) [6, 12]
Both ends included (closed)
Answer: {x : x β β, 6 β€ x β€ 12}
Both ends included (closed)
Answer: {x : x β β, 6 β€ x β€ 12}
(iii) (6, 12]
6 not included, 12 included
Answer: {x : x β β, 6 < x β€ 12}
6 not included, 12 included
Answer: {x : x β β, 6 < x β€ 12}
(iv) [-23, 5)
-23 included, 5 not included
Answer: {x : x β β, -23 β€ x < 5}
-23 included, 5 not included
Answer: {x : x β β, -23 β€ x < 5}
(i) {x : x β β, -3 < x < 0}
(ii) {x : x β β, 6 β€ x β€ 12}
(iii) {x : x β β, 6 < x β€ 12}
(iv) {x : x β β, -23 β€ x < 5}
(ii) {x : x β β, 6 β€ x β€ 12}
(iii) {x : x β β, 6 < x β€ 12}
(iv) {x : x β β, -23 β€ x < 5}
8
What universal set(s) would you propose for each?
(i) The set of right triangles
(ii) The set of isosceles triangles
π‘ Click to View Solution
1 Understanding Universal Set
Universal set U contains ALL elements we're considering
Every other set should be a subset of U
Every other set should be a subset of U
2 Solution
(i) Right triangles
Right triangles are special types of triangles
Suitable universal set: The set of all triangles
Right triangles are special types of triangles
Suitable universal set: The set of all triangles
(ii) Isosceles triangles
Isosceles triangles (two equal sides) are also triangles
Suitable universal set: The set of all triangles
Isosceles triangles (two equal sides) are also triangles
Suitable universal set: The set of all triangles
For both (i) and (ii): The set of all triangles
9
Which could be universal set(s) for all three sets A, B, C?
A = {1, 3, 5}, B = {2, 4, 6}, C = {0, 2, 4, 6, 8}
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) Ο
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
π‘ Click to View Solution
1 Requirement for universal set
For U to be a universal set for A, B, C:
A β U AND B β U AND C β U
All elements of A, B, C must be in U
A β U AND B β U AND C β U
All elements of A, B, C must be in U
2 Check each option
(i) {0, 1, 2, 3, 4, 5, 6}
A = {1, 3, 5} β? β YES (1, 3, 5 are present)
B = {2, 4, 6} β? β YES (2, 4, 6 are present)
C = {0, 2, 4, 6, 8} β? β NO! (8 is NOT in this set)
NOT a universal set β
A = {1, 3, 5} β? β YES (1, 3, 5 are present)
B = {2, 4, 6} β? β YES (2, 4, 6 are present)
C = {0, 2, 4, 6, 8} β? β NO! (8 is NOT in this set)
NOT a universal set β
(ii) Ο (empty set)
Can A, B, C be subsets of empty set? NO!
A has elements like 1, 3, 5
NOT a universal set β
Can A, B, C be subsets of empty set? NO!
A has elements like 1, 3, 5
NOT a universal set β
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 3, 5} β? β YES
B = {2, 4, 6} β? β YES
C = {0, 2, 4, 6, 8} β? β YES
YES, this is a universal set β
A = {1, 3, 5} β? β YES
B = {2, 4, 6} β? β YES
C = {0, 2, 4, 6, 8} β? β YES
YES, this is a universal set β
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 3, 5} β? β YES
B = {2, 4, 6} β? β YES
C = {0, 2, 4, 6, 8} β? β NO! (0 is NOT in this set)
NOT a universal set β
A = {1, 3, 5} β? β YES
B = {2, 4, 6} β? β YES
C = {0, 2, 4, 6, 8} β? β NO! (0 is NOT in this set)
NOT a universal set β
Only (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a suitable universal set
π‘ Key: Universal set must contain ALL elements from A, B, and C (and possibly more)
