π Exercise 1.2 - Sets
Null Sets, Finite & Infinite Sets, and Equal Sets - Complete Solutions
1
Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x : x is a natural number, x < 5 and x > 7}
(iv) {y : y is a point common to any two parallel lines}
π‘ Click to View Solution
1 Understanding Null Set
A null set (or empty set) is a set with NO elements.
Notation: Ο or { }
It has zero members.
Notation: Ο or { }
It has zero members.
2 Analyze each set
(i) Set of odd natural numbers divisible by 2:
Can an odd number be divisible by 2? NO!
Odd numbers are: 1, 3, 5, 7, 9, ... (never divisible by 2)
This is a NULL SET β
Can an odd number be divisible by 2? NO!
Odd numbers are: 1, 3, 5, 7, 9, ... (never divisible by 2)
This is a NULL SET β
(ii) Set of even prime numbers:
Prime numbers: 2, 3, 5, 7, 11, 13, ...
Even prime numbers: Only 2 is even and prime!
Set = {2} β Ο
NOT a null set β (This is a SINGLETON SET)
Prime numbers: 2, 3, 5, 7, 11, 13, ...
Even prime numbers: Only 2 is even and prime!
Set = {2} β Ο
NOT a null set β (This is a SINGLETON SET)
(iii) {x : x < 5 and x > 7}:
Can a number be less than 5 AND greater than 7? IMPOSSIBLE!
There is no natural number satisfying both conditions.
This is a NULL SET β
Can a number be less than 5 AND greater than 7? IMPOSSIBLE!
There is no natural number satisfying both conditions.
This is a NULL SET β
(iv) {y : y is common point to two parallel lines}:
Do two parallel lines intersect? NO!
Parallel lines never meet, so no common points exist.
This is a NULL SET β
Do two parallel lines intersect? NO!
Parallel lines never meet, so no common points exist.
This is a NULL SET β
β NULL SETS: (i), (iii), (iv)
β NOT NULL SET: (ii) - This is a SINGLETON SET = {2}
β NOT NULL SET: (ii) - This is a SINGLETON SET = {2}
π Note: A natural number > 1 is prime if it has only two divisors: 1 and itself.
Prime numbers: {2, 3, 5, 7, 11, 13, ...}
Prime numbers: {2, 3, 5, 7, 11, 13, ...}
2
Which of the following sets are finite or infinite?
(i) The set of months of a year
(ii) {1, 2, 3, ...}
(iii) {1, 2, 3, ..., 99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
π‘ Click to View Solution
1 Understanding Finite & Infinite Sets
Finite Set: Has a definite (countable) number of elements.
Example: {1, 2, 3, 4, 5} - we can count them all
Infinite Set: Has unlimited elements - counting never ends.
Example: {1, 2, 3, 4, ...} - the "..." means it continues forever
Example: {1, 2, 3, 4, 5} - we can count them all
Infinite Set: Has unlimited elements - counting never ends.
Example: {1, 2, 3, 4, ...} - the "..." means it continues forever
2 Analyze each set
(i) Set of months of a year:
How many months? Exactly 12 months
We can count them: Jan, Feb, Mar, ..., Dec
FINITE SET β
How many months? Exactly 12 months
We can count them: Jan, Feb, Mar, ..., Dec
FINITE SET β
(ii) {1, 2, 3, ...}:
The dots "..." mean it continues forever!
We can never count all the numbers.
INFINITE SET β
The dots "..." mean it continues forever!
We can never count all the numbers.
INFINITE SET β
(iii) {1, 2, 3, ..., 99, 100}:
Starts at 1 and ends at 100
Total elements: exactly 100
FINITE SET β
Starts at 1 and ends at 100
Total elements: exactly 100
FINITE SET β
(iv) Positive integers greater than 100:
Set includes: 101, 102, 103, 104, ...
There's no end! We can never count them all.
INFINITE SET β
Set includes: 101, 102, 103, 104, ...
There's no end! We can never count them all.
INFINITE SET β
(v) Prime numbers less than 99:
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
It has a DEFINITE COUNT (finite number)
FINITE SET β
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
It has a DEFINITE COUNT (finite number)
FINITE SET β
FINITE: (i), (iii), (v)
INFINITE: (ii), (iv)
INFINITE: (ii), (iv)
3
State whether each of the following sets is finite or infinite
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)
π‘ Click to View Solution
1 Solution for each
(i) Lines parallel to x-axis:
There are infinitely many lines we can draw parallel to the x-axis.
We can draw them above, below, or at any height!
INFINITE SET β
There are infinitely many lines we can draw parallel to the x-axis.
We can draw them above, below, or at any height!
INFINITE SET β
(ii) Letters in English alphabet:
English has exactly 26 letters: A, B, C, ..., Z
We can count them all.
FINITE SET β
English has exactly 26 letters: A, B, C, ..., Z
We can count them all.
FINITE SET β
(iii) Multiples of 5:
Set: {5, 10, 15, 20, 25, 30, ...}
We can keep multiplying by 5 forever!
INFINITE SET β
Set: {5, 10, 15, 20, 25, 30, ...}
We can keep multiplying by 5 forever!
INFINITE SET β
(iv) Animals living on earth:
We can count all animals (though it's a huge number)
There's a definite, finite number of animals on earth
FINITE SET β
We can count all animals (though it's a huge number)
There's a definite, finite number of animals on earth
FINITE SET β
(v) Circles passing through origin (0, 0):
We can draw circles of different sizes through (0, 0)
There's no limit to how many circles we can draw!
INFINITE SET β
We can draw circles of different sizes through (0, 0)
There's no limit to how many circles we can draw!
INFINITE SET β
FINITE: (ii), (iv)
INFINITE: (i), (iii), (v)
INFINITE: (i), (iii), (v)
4
State whether A = B or not
(i) A = {a, b, c, d}, B = {d, c, b, a}
(ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}, B = {x : x is positive even integer and x β€ 10}
π‘ Click to View Solution
1 What makes sets equal?
Two sets are EQUAL (A = B) if they have:
β Exactly the same elements
β Same number of elements
β ORDER DOESN'T MATTER! {1, 2, 3} = {3, 2, 1}
Sets are NOT EQUAL (A β B) if:
β Even one element differs
β Exactly the same elements
β Same number of elements
β ORDER DOESN'T MATTER! {1, 2, 3} = {3, 2, 1}
Sets are NOT EQUAL (A β B) if:
β Even one element differs
2 Analyze each pair
(i) A = {a, b, c, d}, B = {d, c, b, a}
A has: a, b, c, d
B has: d, c, b, a (same elements, just different order!)
Both have 4 elements, same elements
A = B β
A has: a, b, c, d
B has: d, c, b, a (same elements, just different order!)
Both have 4 elements, same elements
A = B β
(ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
A has: 4, 8, 12, 16
B has: 8, 4, 16, 18
A has 12, but B has 18 instead
12 β A but 12 β B
A β B β
A has: 4, 8, 12, 16
B has: 8, 4, 16, 18
A has 12, but B has 18 instead
12 β A but 12 β B
A β B β
(iii) A = {2, 4, 6, 8, 10}, B = {x : x is positive even integer and x β€ 10}
First, convert B to roster form:
Positive even integers β€ 10: 2, 4, 6, 8, 10
B = {2, 4, 6, 8, 10}
A and B have exactly the same elements!
A = B β
First, convert B to roster form:
Positive even integers β€ 10: 2, 4, 6, 8, 10
B = {2, 4, 6, 8, 10}
A and B have exactly the same elements!
A = B β
(i) A = B
(ii) A β B
(iii) A = B
(ii) A β B
(iii) A = B
5
Are the following pairs of sets equal? Give reasons
(i) A = {2, 3}, B = {x : x is solution of xΒ² + 5x + 6 = 0}
(ii) A = {x : x is a letter in the word FOLLOW}, B = {y : y is a letter in the word WOLF}
π‘ Click to View Solution
1 Part (i) - Solve the equation
A = {2, 3}
For B: Solve xΒ² + 5x + 6 = 0
Factorize:
xΒ² + 5x + 6 = xΒ² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3) = 0
Solutions: x = -2 or x = -3
B = {-2, -3}
A = {2, 3} but B = {-2, -3}
2 β A but 2 β B
A β B β
For B: Solve xΒ² + 5x + 6 = 0
Factorize:
xΒ² + 5x + 6 = xΒ² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3) = 0
Solutions: x = -2 or x = -3
B = {-2, -3}
A = {2, 3} but B = {-2, -3}
2 β A but 2 β B
A β B β
2 Part (ii) - List letters
A = letters in FOLLOW
Letters: F, O, L, L, O, W
Drop repetitions: F, O, L, W
A = {F, O, L, W}
B = letters in WOLF
Letters: W, O, L, F
(No repetitions)
B = {W, O, L, F}
A = {F, O, L, W} and B = {W, O, L, F}
Same elements, different order!
A = B β
Letters: F, O, L, L, O, W
Drop repetitions: F, O, L, W
A = {F, O, L, W}
B = letters in WOLF
Letters: W, O, L, F
(No repetitions)
B = {W, O, L, F}
A = {F, O, L, W} and B = {W, O, L, F}
Same elements, different order!
A = B β
(i) A β B (because A = {2, 3} but B = {-2, -3})
(ii) A = B (same letters, just different order)
(ii) A = B (same letters, just different order)
6
From the sets given below, select equal sets
A = {2, 4, 8, 12}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {-1, 1}
F = {0, a}
G = {1, -1}
H = {0, 1}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {-1, 1}
F = {0, a}
G = {1, -1}
H = {0, 1}
π‘ Click to View Solution
1 Compare each pair
Let me check which sets have the exact same elements (order doesn't matter):
Checking B and D:
B = {1, 2, 3, 4}
D = {3, 1, 4, 2} = {1, 2, 3, 4} (rearranged)
β B = D
Checking E and G:
E = {-1, 1}
G = {1, -1} = {-1, 1} (rearranged)
β E = G
Checking others:
A = {2, 4, 8, 12} β No match
C = {4, 8, 12, 14} β No match
F = {0, a} β No match (has unknown 'a')
H = {0, 1} β No match
Checking B and D:
B = {1, 2, 3, 4}
D = {3, 1, 4, 2} = {1, 2, 3, 4} (rearranged)
β B = D
Checking E and G:
E = {-1, 1}
G = {1, -1} = {-1, 1} (rearranged)
β E = G
Checking others:
A = {2, 4, 8, 12} β No match
C = {4, 8, 12, 14} β No match
F = {0, a} β No match (has unknown 'a')
H = {0, 1} β No match
2 Verify results
B = {1, 2, 3, 4}
D = {3, 1, 4, 2}
Rearranging D: 1, 2, 3, 4 = Same as B β
E = {-1, 1}
G = {1, -1}
Rearranging G: -1, 1 = Same as E β
D = {3, 1, 4, 2}
Rearranging D: 1, 2, 3, 4 = Same as B β
E = {-1, 1}
G = {1, -1}
Rearranging G: -1, 1 = Same as E β
β B = D
β E = G
β E = G
π‘ Remember: Order of elements doesn't matter in sets! {1, 2} = {2, 1}
