Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 10 Sets

IMPORTANT POINTS

In our day-to-day life we often speak or hear about different types of collections.
Such as :
(i) A collection of stamps.
(ii) A collection of toys :
(iii) A collection of books, etc.
In the same way, we have different types of groups made for different activities.
Such as :
(i) A group of boys playing hockey.
(ii) A group of girls playing badminton.
(iii) A group of students going for picnic, etc.
In mathematics, a collection of particular things or a group of particular objects is called a set.

1. Definition of a Set : A set is a collection of well-defined objects.

2. Meaning of “Well-Defined” : Well- defined means, it must be absolutely clear that which object belongs to the set and which does not.

3. Elements (or, members) of a set : The objects used to form a set are called its elements or its members.
Keep in Mind :

• The pair of curly braces { } denotes a set.
• The Greek letter Epsilon ‘ ε ’ is used for the words “belongs to”, “is an element of, etc.

p ε A will be be read as “p belongs to set A” or “p is an element of set A”.
In the same way ; q ε A, r ε A, s ε a. and t ε A.
The symbol ‘ ε not ’ stands for “does not belong to” also for “is not an element of.
a ε A will be read as “a does not belong to set A” or “a is not an element of set A”.

4. Properties of a set :

• The change in order of writing the elements does not make any change in the set.
• If one or many elements of a set are repeated, the set remains the same.

5. Notation (Representation) of a set :

• Description method : In this method, a well-defined description about the elements of the set is made.
• Roster or Tabular Method : In this method, the elements (members) of a set are written inside a pair of curly braces and are separated by commas.
• Rule or Set Builder Method : In this method, actual elements of the set are not listed but a rule or a statement or a formula, in the briefest possible way, is written inside a pair of curly braces.

EXERCISE 10(A)

Question 1: State whether or not the following elements form a set; if not, give reason:
(i) All easy problems in your text book.
(ii) All three sided figures.
(iii) The first five counting numbers.
(iv) All the tall boys of your class.
(v) The last three days of the week.
(vi) All triangles that are difficult to draw.
(vii) The first three letters of the English alphabet.
(viii) All tasty fruits.
(ix) All clever boys of class 6.
(x) All good schools in Delhi.
(xi) All the girls in your class, whose heights are less than your height.
(xii) All the boys in your class, whose heights are more than your height.
(xiii) All the problems in your Mathematics book, which are difficult for Amit.

Answer:
(i) No; some problems may be easy for one person but may be difficult to some other person. Objects are not well- defined.
(ii) Yes.
(iii) Yes.
(iv) No; it is not mentioned that the boys must be taller than which boy. If we consider three boys A, B and C; boy B can be taller than A but not necessarily taller than C.
(v) Yes
(vi) No; it may be difficult for one boy to draw a given triangle but to some other boy it may be easy to draw the same triangle.
(vii) Yes
(viii) No; a fruit may be tasty for one person and may not be tasty to other person / persons.
(ix) No; clever in what respect and from whom out of six ?
(x) No; all the people can not find the same schools as good as others said. So, the objects are not well-defined.
(xi) Yes
(xii) Yes
(xiii) Yes.
A set must be based on facts that everyone agrees on, rather than personal feelings. When terms like "easy" or "tasty" are used, the criteria change from person to person, making it not well-defined.
Teacher's Tip: If a collection involves an opinion or an adjective like "good," "bad," or "tall," it is usually not a set.
Exam Tip: For "well-defined" questions, always ask yourself if every person in the world would give the exact same list of items.

EXERCISE 10(B)

Question 1: If set A = {2, 3, 4, 5, 6} , state which of the following statements arc true and which are false :
(i) 2 ε A
(ii) 5, 6 ε A
(iii) 3, 4, 7 ε A
(iv) 2, 8 ε A

Answer:
(i) True
(ii) True
(iii) False
(iv) False
The symbol ε means the number is a member of the specific group mentioned. In these problems, we check if every number listed before the symbol is actually present inside the curly braces of set A.
Teacher's Tip: Think of the set like a box; if the item is inside the box, the statement "belongs to" is true.
Exam Tip: If a statement lists multiple elements like "3, 4, 7," all of them must be in the set for the statement to be true.

Question 2: If set B = {4, 6, 8, 10, 12, 14} . State, which of the following statements is correct and which is wrong :
(i) 5 ε B
(ii) 12 ε B
(iii) 14 ε B
(iv) 9 ε B
(v) B is a set of even numbers between 2 and 16.
(vi) 4, 6 and 10 are the members of the set B. Also, write the wrong statements correctly.

Answer:
(i) Wrong ; 5 ∉ B
(ii) Correct
(iii) Correct
(iv) Wrong ; 9 ∉ B
(v) Correct
(vi) Correct.
To correct a wrong membership statement, we change the "belongs to" symbol to "does not belong to." This accurately reflects that numbers like 5 and 9 are missing from the list in set B.
Teacher's Tip: Use the slash through the epsilon ∉ to show that an element is an outsider to the set.
Exam Tip: When writing a set description, ensure the boundaries (like "between 2 and 16") perfectly match the numbers listed.

Question 3: State, whether true or false :
(i) Sets {4, 9, 6, 2} and {6, 2, 4, 9} are not the same.
(ii) Sets {0, 1, 3, 9, 4} and {4, 0, 1, 3, 9} are the same.
(iii) Sets {5, 4} and {5, 4, 4, 5} are not the same.
(iv) Sets {8, 3} and {3, 3, 8} are the same.
(v) Collection of vowels used in the word ‘ALLAHABAD’ forms a set.
(vi) If P is the set of letters in the word ‘ROOP’; then P = {p, o, r}
(vii) If M is the set of letters used in the word ‘MUMBAI’, then: M = {m, u, b, a, i}

Answer:
(i) False.
(ii) True.
(iii) False.
(iv) True.
(v) True.
(vi) True.
(vii) True.
In sets, the order in which elements are written does not change the identity of the set. Furthermore, repeating an element does not create a new set; {5, 4} is exactly equal to {5, 4, 4, 5} .
Teacher's Tip: Think of a set like a team roster; it doesn't matter who you name first, the team remains the same.
Exam Tip: When listing letters from a word to form a set, remember to only write each unique letter once.

Question 4: Write the set containing :
(i) the first five counting numbers.
(ii) the three types of angles.
(iii) the three types of triangles.
(iv) the members of your family.
(v) the first six consonants of the English Alphabet.
(vi) the first four vowels of the English Alphabet.
(vii) the names of any three Prime-Ministers of India.

Answer:
(i) {1, 2, 3, 4, 5}
(ii) {acute-angle, obtuse-angle, right-angle} .
(iii) {scalene triangle, isosceles triangles, equilateral triangle} .
(iv) { Write the name of family member}  .
(v) {b, c, d, f, g, h}
(vi) {a, e, i, o}
(vii) {J.L. Nehru, A.B. Vajpayee, Dr. Manmohan Singh} .
These answers use the Roster method where individual members are listed within curly brackets. We ensure all criteria, such as "first five" or "first six consonants," are strictly followed.
Teacher's Tip: For consonants, always skip the vowels (a, e, i, o, u) when counting your list.
Exam Tip: Always use commas to separate elements inside the curly braces to avoid losing marks for formatting.

Question 5:
(a) Write the members (elements) of each set given below :
(i) {3, 8, 5, 15, 12, 7}
(ii) {c, m, n, o, s}
(b) Write the sets whose elements are :
(i) 2, 4, 8, 16, 64 and 128
(ii) 3, 5, 15, 45, 75 and 90

Answer:
(a) (i) 3, 8, 5, 15, 12 and 7
(ii) c, m, n, o and s
(b) (i) {2, 4, 8, 16, 64, 128}
(ii) {3, 5, 15, 45, 75, 90}
Listing elements means removing the brackets and just showing the items. Creating a set from elements means placing those items inside curly braces.
Teacher's Tip: Elements are the "stuff" inside the set, while the set is the "container" itself.
Exam Tip: If the question asks for "elements," do not use curly braces; if it asks for "sets," you must use them.

Question 6:
(i) Write the set of letters used in the word ‘BHOPAL’.
(ii) Write the set of vowels used in the word ‘BENGAL’.
(iii) Write the set of consonants used in the word ‘HONG KONG’.

Answer:
(i) {b, h, o, p, a, l}
(ii) {e, a}
(iii) {h, o, n, g, k}
Note: In (iii), 'o' is technically a vowel, but the provided answer lists it. Usually, consonants are letters other than a, e, i, o, u. For (i), every letter is unique so we list them all.
Teacher's Tip: When extracting letters from a word, cross them out as you write them so you don't repeat any.
Exam Tip: Double-check if the question asks specifically for vowels, consonants, or all letters.

EXERCISE 10(C)

Question 1: Write each of the following sets in the Roster Form :
(i) The set of five numbers each of which is divisible by 3.
(ii) The set of integers between -4 and 4.
(iii) {x: x { is a letter in the word 'SCHOOL'}}
(iv) {x: x { is an odd natural number between 10 and 20}}
(v) {Vowels used in the word 'AMERICA'}}
(vi) {Consonants used in the word 'MADRAS'}}

Answer:
(i) {3, 6, 9, 12, 15}
(ii) {-3, -2, -1, 0, 1, 2, 3}
(iii) {s, c, h, o, l}
(iv) {11, 13, 15, 17, 19}
(v) {a, e, i}
(vi) {m, d, r, s}
Roster form requires listing every specific element that fits the rule. For example, "between -4 and 4" means we start at -3 and end at 3.
Teacher's Tip: "Between" means you do not include the starting and ending numbers themselves.
Exam Tip: In word problems like "SCHOOL," remember that 'O' appears twice but is written only once in the set.

Question 2: Write each given set in the Roster Form :
(i) All prime numbers between one and twenty.
(ii) The squares of first four natural numbers.
(iii) Even numbers between 1 and 9.
(iv) First eight letters of the English alphabet.
(v) The letters of the word ‘BASKET’.
(vi) Four cities of India whose names start with the letter J.
(vii) Any four closed geometrical figures.
(viii) Vowels used in the word ‘MONDAY’.
(ix) Single digit numbers that are perfect squares as well.

Answer:
(i) {2, 3, 5, 7, 11, 13, 17, 19}
(ii) {1², 2², 3², 4²} = {1, 4, 9, 16}
(iii) {2, 4, 6, 8}
(iv) {a, b, c, d, e, f, g, h}
(v) {b, a, s, k, e, t}
(vi) {Jaipur, Jodhpur, Jalandhar, Jaunpur}
(vii) {triangle, bigcirc, square, {hexagon}}
(viii) {o, a}
(ix) {0, 1, 4, 9}
These examples show how mathematical properties like "prime" or "perfect square" define which numbers enter the set. We carefully list only those that satisfy every part of the description.
Teacher's Tip: 1 is neither prime nor composite, so never include it in a set of prime numbers.
Exam Tip: For "perfect squares," remember that 0² = 0 , 1² = 1 , 2² = 4 , and 3² = 9 are all single digits.

Question 3: Write each given set in the Set- Builder Form :
(i) {2, 4, 6, 8, 10}
(ii) {2, 3, 5, 7, 11}
(iii) {January, June, July}
(iv) {a, e, i, o, u}
(v) {Tuesday, Thursday}
(vi) {1, 4, 9, 16, 25}
(vii) {5, 10, 15, 20, 25, 30}

Answer:
(i) {x : x is an even natural number less than 12}
(ii) {x : x is a prime number less than 12}
(iii) {x : x is a months of the year whose name starts with letter J}
(iv) {x : x is a vowel in English alphabets}
(v) {x : x is a day of the week whose name starts with letter T}
(vi) {x : x is a perfect square natural number upto 25}
(vii) {x : x  is a natural number upto 30 and divisible by 5} .
Set-Builder form uses a variable x and a colon to describe the common property of all elements. It explains "what" the elements are rather than just listing them.
Teacher's Tip: Read the colon : as "such that" when you are writing or speaking Set-Builder notation.
Exam Tip: Make sure your description is specific enough so that someone else would list exactly the same elements.

Question 4: Write each of the following sets in Roster (tabular) Form and also in Set-Builder Form.
(i) Set of all natural numbers that can divide 24 completely.
(ii) Set of odd numbers between 20 and 35.
(iii) Set of letters used in the word ‘CALCUTTA’.
(iv) Set of names of the first five months of a year.
(v) Set of all two digit numbers that are perfect square as well.

Answer:
(i) {1, 2, 3, 4, 6, 8, 12, 24} ; {x : x is a natural number which divides 24 completely}
(ii) {21, 23, 25, 27, 29, 31, 33} ; {x: x is an odd number between 20 and 35}
(iii) {c, a, l, u, t} ; {x: x is a letter used in the word ‘CALCUTTA’}
(iv) {January, February, March, April, May} ; {x : x is name of first five months of a year}
(v) {16, 25, 36, 49, 64, 81} ; {x : x is a perfect square two digit number}.
This question demonstrates how to switch between specific lists and general rules. For divisors of 24, we find every factor, and for odd numbers, we skip all even values in the range.
Teacher's Tip: For perfect squares, think of 4², 5², ..... until you hit a three-digit number ( 10² = 100 ).
Exam Tip: Ensure that your Roster form elements are separated by commas and enclosed in curly braces.

Question 5: Write, in Roster Form, the set of :
(i) the first four odd natural numbers each divisible by 5.
(ii) the counting numbers between 15 and 35; each of which is divisible by 6.
(iii) the names of the last three days of a week.
(iv) the names of the last four months of a year.

Answer:
(i) {5, 15, 25, 35}
(ii) {18, 24, 30}
(iii) {Friday, Saturday, Sunday}
(iv) {September, October, November, December} .
We filter the numbers or names based on two conditions. For example, in (i), the numbers must be both odd and multiples of 5.
Teacher's Tip: To find odd numbers divisible by 5, just look at the multiples of 5 that end in the digit 5.
Exam Tip: Be careful with boundaries; "between 15 and 35" means 15 and 35 are not part of the search.

EXERCISE 10(D)

Question 1: 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}
(v) {7, 14, 21, ....., 2401}

Answer:
(i) Infinite
(ii) Finite
(iii) Infinite
(iv) Finite
(v) Finite
A set is finite if its elements can be counted and have a definitive end. A set is infinite if it continues without stopping, usually indicated by dots that don't end in a final number.
Teacher's Tip: If you see dots .... at the very end of the set with no number after them, it's usually infinite.
Exam Tip: Even if a set is very large (like 1 to 1,000,000), it is still "finite" because it has an end.

Question 2:
(i) Which of the following sets is empty?
(ii) Set of counting numbers between 5 and 6.
(iii) Set of odd numbers between 7 and 19. Set of odd numbers between 7 and 9.
(iv) Set of even numbers which are not divisible by 2.
(v) {0} .

Answer:
(i), (iii) (the second part) and (iv) are empty.
An empty set contains absolutely no elements. For example, there is no whole counting number between 5 and 6, so that collection is empty.
Teacher's Tip: Don't confuse {0} with an empty set; {0} has one member (zero) while an empty set has none.
Exam Tip: Use the symbol phi or empty braces { } to represent an empty set in your answers.

Question 3: State, which pair of sets, given below, are equal sets or equivalent sets:
(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}
(v) {5, 5, 2, 4} and {5, 4, 2, 2}

Answer:
(i) Equal
(ii) Equivalent
(iii) Equal
(iv) Equivalent
(v) Equal
Equal sets have exactly the same items, regardless of order or repetition. Equivalent sets only need to have the same number of items (count), even if the items themselves are different.
Teacher's Tip: Remember: All equal sets are equivalent, but not all equivalent sets are equal.
Exam Tip: Count the elements carefully; if the counts match, write "Equivalent." If the content matches too, write "Equal."

Question 4: State, which of the following are finite or infinite sets :(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) Infinite
(ii) Infinite
(iii) Infinite
(iv) Finite
(v) Finite
(vi) Finite
(vii) Finite
(viii) Infinite
(ix) Infinite
(x) Infinite
Some sets, like the number of people in a country, are very large but still countable, making them finite. Others, like the points on a line, can be divided forever, making them infinite.
Teacher's Tip: If you can eventually finish counting it (even if it takes years), it is finite.
Exam Tip: Be careful with math concepts; there are infinite numbers (fractions) between any two whole numbers.

Question 5: State, whether or not the following sets are empty:
(i) {Prime numbers divisible by 2}
(ii) {Negative natural numbers}
(iii) {Women with height 5 metre}
(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, which are more than 9 and less than 15.

Answer:
(i) Not empty
(ii) Empty
(iii) Empty
(iv) Not empty
(v) Not empty
(vi) Empty
(vii) Not empty
A set is not empty if you can find even one thing that belongs in it. For example, 2 is a prime number divisible by 2, so that set contains the number 2.
Teacher's Tip: Natural numbers start from 1, so there is no such thing as a "negative" natural number.
Exam Tip: Always look for the single exception (like the number 2) before calling a set empty.

Question 6: 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) Equivalent
(ii) Equal
(iii) Equal
(iv) Equal
(v) Equal
(vi) Equivalent
For words like "MEMBER" and "REMEMBER," the set of unique letters is {m, e, b, r} for both, making them equal. In (v), both descriptions describe empty sets, and all empty sets are considered equal.
Teacher's Tip: When comparing words, write out the unique letters first to see if they match exactly.
Exam Tip: Sets of infinite things (like all evens vs. all odds) are often considered equivalent because they follow the same pattern of growth.

Question 7: 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) Finite
(ii) Infinite
(iii) Finite
(iv) Finite
(v) Infinite
(vi) Finite
A set that starts with ..... but has no beginning point is infinite towards the negative side. Similarly, a set of whole numbers "greater than 20" goes on to billions and beyond without ever stopping.
Teacher's Tip: "Greater than" usually leads to an infinite set unless there is a maximum cap mentioned.
Exam Tip: Check both ends of the curly braces for dots to decide if the set is infinite.

Question 8: 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, dots is a finite set.

Answer:
(i) False
(ii) False
(iii) True
(iv) True (each set is the empty set)
(v) True
(vi) False (6 is an even natural number which is divisible by 3)
(vii) True (no positive number can be less than 0)
(viii) False
Statement (i) is false because the dots at the start mean it continues infinitely in the negative direction. For (vi), it is false because 6, 12, 18 etc. are even and divisible by 3.
Teacher's Tip: If you can find just one counter-example (like the number 6), the "empty set" claim is immediately false.
Exam Tip: Read "True or False" questions very slowly to catch small details like "positive" vs "negative."

Question 9: 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 multiples of 3 and less than 30} and B = {Natural numbers divisible by 4 and between 20 and 45}
(vi) P = {Letters in the word ‘ALLAHABAD’} and Q = {Letters in the word ‘MUSSOORIE’}

Answer:
(i) Disjoint sets; as no girl can be of age below 15 years and also above 15 years
(ii) Overlapping sets; as boys above 27 years are also above 20 years.
(iii) Overlapping sets; as natural numbers from 50 to 59 are common to both the sets.
(iv) Overlapping sets; as students of class IX studying in I,C.S.E. board are common.
(v) Overlapping sets; as natural number 24 is common to both the sets.
(vi) Disjoint sets; as no letter is common to both the sets.
Disjoint sets have zero members in common, like being both above and below 15 at the same time. Overlapping sets have at least one member that belongs to both groups.
Teacher's Tip: Overlapping sets are like a Venn diagram with a shared middle section; disjoint sets are two separate circles.
Exam Tip: To prove a set is overlapping, you only need to find one single element that appears in both.

EXERCISE 10(E)

Question 1: Write the cardinal number of each of the following sets :(i) A = {0, 1, 2, 4}
(ii) B = {-3, -1, 1, 3, 5, 7}
(iii) C = { }
(iv) D = {3, 2, 2, 1, 3, 1, 2}
(v) E = {Natural numbers between 15 and 20}
(vi) F = {Whole numbers from 8 to 14} .

Answer:
(i) A = {0, 1, 2, 4} i.e. n (A) = 4
(ii) B = {-3, -1, 1, 3, 5, 7} i.e. n(B) = 6
(iii) C = { } i.e. n(C) = 0
(iv) D = {3, 2, 2, 1, 3, 1, 2} , D = {3, 2, 1} i.e. n(D) = 3
(v) E = {16, 17, 18, 19} i.e. n(E) = 4
(vi) F = {8, 9, 10, 11, 12, 13, 14} i.e. n(F) = 7
The cardinal number, written as n(A) , is the total count of distinct elements in the set. If an element like "2" or "3" is repeated, we only count it once.
Teacher's Tip: Think of the cardinal number as the "answer" to the question "How many items are inside?"
Exam Tip: Always rewrite a set without repetitions before you start counting its cardinal number.

Question 2: Given:
(i) A = {Natural numbers less than 10}
B = {Letters of the word ‘PUPPET’}
C = {Squares of first four whole numbers}
D = {Odd numbers divisible by 2}. Find:
(i) n(A)
(ii) n(B)
(iii) n(C)
(iv) n(D) (v) A ∪ B and n(A ∪ B)
(vi) A ∩ C and n(A ∩ C)
(vii) n(B ∪ D)
(viii) n(C ∩ D)
(ix) n(B ∪ C)
(x) n(A ∪ D) .

Answer:
Here,

A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {P, U, E, T}
C = {0, 1, 4, 9}
D = { } or phi
(i) n(A) = 9
(ii) n(B) = 4
(iii) n(C) = 4
(iv) n(D) = 0
(v) A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, P, U, E, T} and n(A ∪ B) = 13
(vi) A ∩ C = {1, 4, 9} and n(A ∩ C) = 3
(vii) B ∪ D = {P, U, E, T} , n(B ∪ D) = 4
(viii) C ∩ D = { } , n(C ∩ D) = 0
(ix) B ∪ C = {P, U, E, T, 0, 1, 4, 9} , n(B ∪ C) = 8
(x) A ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9} , n(A ∪ D) = 9
Union ( ∪ ) means combining everything from both sets, while Intersection ( ∩ ) means taking only what they share. Since set D is empty, uniting it with any set doesn't change the original count.
Teacher's Tip: Use the letter "U" in Union to remember it means "United" (everyone included).
Exam Tip: When doing A ∪ B , write all elements of A first, then add elements of B that aren't already there.

Question 3: State true or false for each of the following. Correct the wrong statement.
(i) If A = {0} , then n(A) = 0 .
(ii) n(phi) = 1 .
(iii) If T = {a, l, a, h, b, d, h} , then n(T) = 5 .
(iv) If B = {1, 5, 51, 15, 5, 1} , then n(B) = 6 .

Answer:
(i) False. True statement is n(A) = 1
(ii) False. i.e. n(phi) = 0
(iii) True. T = {a, l, h, b, d} i.e. n(T) = 5
(iv) False. B = {1, 5, 51, 15} , n(B) = 4
These problems reinforce that the cardinal number depends on the unique count of items. Even if "1" and "5" appear twice in set B, they only represent two unique members of the collection.
Teacher's Tip:  phi is a name for the empty set; it has zero items, so its count is always 0.
Exam Tip: Never count repeated items when calculating n(A) , or you will get the answer wrong.