Unit notes and practice questions

Grade 9 Mathematics unit 1: Further on Sets

Everything here is for unit 1 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

3
Cards: 10
Practice questions on unit 1: 1–16
The textbook pages for unit 1

What unit 1 covers

1.Description of the Concept Set
2.Notion of Set
3.Operations on Sets
4.Application of Operations on Sets

Short notes for unit 1

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

Set counting formulasReference tableTwo sets · Complement · In A only · +4

Rule	Formula
Two sets	$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Complement	$n(A') = n(U) - n(A)$
In $A$ only	$n(A) - n(A \cap B)$
Three sets	$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$
Disjoint sets	$n(A \cup B) = n(A) + n(B)$
De Morgan	$(A \cup B)' = A' \cap B'$
De Morgan	$(A \cap B)' = A' \cup B'$
Subsets of a set	$2^{n}$ for a set of $n$ members

When you use it

Reach for these when a question counts people or things that belong to more than one group at once, which is what every Venn diagram question really is. Draw the diagram, fill the innermost region first, and work outwards.

Watch out

With three sets the signs alternate: subtract each pair, then add the triple back. Leaving that last term out is the single most common way a three-circle Venn question goes wrong, because the middle region ends up counted zero times instead of once.

Cardinality of Union of Two SetsFormula

n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A \cup B) = n(A) + n(B) - n(A \cap B)

$n(A \cup B)$: number of elements in the union of A and B $1$
$n(A)$: number of elements in set A $1$
$n(B)$: number of elements in set B $1$
$n(A \cap B)$: number of elements in the intersection of A and B $1$

When you use it

Use when solving word problems or calculating the number of elements in overlapping sets.

Watch out

Remember to subtract the intersection n(A and B), otherwise shared elements will be counted twice.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Number of Subsets and Proper SubsetsReference tableAll Subsets · Proper Subsets

Set Type	Formula for n Elements	Example for n = 3
All Subsets	$2^n$	$2^3 = 8$
Proper Subsets	$2^n - 1$	$2^3 - 1 = 7$

When you use it

Use to quickly find the total number of subsets or proper subsets for any finite set.

Watch out

A set is never a proper subset of itself, which is why proper subsets have 1 subtracted from the total.

Drafted from Grade 9 Mathematics, pages 1-86, then checked twice before it went up

Practice questions on unit 1

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Which of these is a well-defined set?

A.The beautiful girls in your class
B.The hardworking teachers in a school
C.The natural numbers less than 100
D.The handsome boys in your school

Show the answer and why

Answer: C The natural numbers less than 100

Why

A set is well-defined when, for any object, we can tell whether it is in the set or not. The natural numbers less than 100 can be listed exactly, so we can always decide if a number belongs. Words like beautiful, hardworking and handsome are opinions, so those collections are not well-defined.

Question 2

Let A = {1, 2, 3, 4, 5, 6}. Which symbol goes in the blank?
8 ____ A

A.∈
B.∉
C.⊆
D.⊂

Show the answer and why

Answer: B ∉

Why

The symbol ∈ means belongs to, and ∉ means does not belong to. The number 8 is not in the list {1, 2, 3, 4, 5, 6}, so we write 8 ∉ A.

Question 3

The set of days in a week is a ______ set.

A.finite
B.infinite
C.empty
D.equivalent

Show the answer and why

Answer: A finite

Why

A finite set has a definite, limited number of elements. A week has exactly 7 days, so the set of days in a week is finite.

Question 4

Let A = {1, 2, 3, ..., 10}. Find n(A), the number of elements of A.

A.9
B.10
C.11
D.100

Show the answer and why

Answer: B 10

Why

The set A has the numbers 1 up to 10. Counting them gives 10 elements, so n(A) = 10.

Question 5

Which set is described by the set builder form {x : x ∈ ℕ and x < 11}?

A.{0, 1, 2, ..., 10}
B.{2, 4, 6, 8, 10}
C.{1, 2, 3, ..., 10}
D.{1, 2, 3, ..., 11}

Show the answer and why

Answer: C {1, 2, 3, ..., 10}

Why

Read the rule as: x is a natural number and x is less than 11. Natural numbers start at 1, so the values are 1 to 10. Zero is not a natural number, and 11 is not less than 11.

Question 6

Let A = {1, 2, 3}. How many proper subsets does A have?

Show the answer and why

Answer: D 7

Why

A finite set with n elements has 2ⁿ subsets, and 2ⁿ − 1 proper subsets. Here n = 3, so there are 2³ − 1 = 8 − 1 = 7 proper subsets. The list is {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} and the empty set.

Question 7

Let A = {1, 2, 3, 4} and B = {Red, Blue, Green, Black}. Which statement is true?

A.A and B are equal sets.
B.A is a proper subset of B.
C.A and B are equivalent sets.
D.A is an infinite set.

Show the answer and why

Answer: C A and B are equivalent sets.

Why

Two finite sets are equivalent when they have the same number of elements. Both A and B have 4 elements, so they are equivalent. They are not equal, because equal sets must have exactly the same elements, and these two sets have different elements.

Question 8

Let A = {0, 1, 3, 5, 7} and B = {1, 2, 3, 4, 6, 7}. Find A ∩ B.

A.{1, 3, 7}
B.{0, 5}
C.{2, 4, 6}
D.{1, 2, 3, 4, 5, 6, 7}

Show the answer and why

Answer: A {1, 3, 7}

Why

The intersection A ∩ B is the set of elements that are in both A and B. The numbers 1, 3 and 7 appear in both sets, so A ∩ B = {1, 3, 7}.

Question 9

Let A = {1, 2, 4, 5, 8} and B = {2, 3, 5, 7}. Find the symmetric difference A ∆ B.

A.{2, 5}
B.{1, 4, 8}
C.{1, 3, 4, 7, 8}
D.{1, 2, 3, 4, 5, 7, 8}

Show the answer and why

Answer: C {1, 3, 4, 7, 8}

Why

First find A \ B, the elements in A but not in B: {1, 4, 8}. Then find B \ A, the elements in B but not in A: {3, 7}. The symmetric difference is the union of these two sets, so A ∆ B = {1, 4, 8} ∪ {3, 7} = {1, 3, 4, 7, 8}.

Question 10

Let A = {2, 4, 6} and B = {1, 3}. Which ordered pair is an element of B × A?

A.(2, 1)
B.(1, 2)
C.(6, 6)
D.(3, 3)

Show the answer and why

Answer: B (1, 2)

Why

In B × A, the first number comes from B and the second number comes from A. B × A = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6)}. The pair (1, 2) is in this list. The pair (2, 1) belongs to A × B, not B × A.

The app holds 14 questions on this unit, marked as you go. These are the first 10.

The textbook pages for unit 1

Unit 1 runs from page 1 to page 16 of the Ministry of Education edition.

Ministry of Education edition, English, 337 pages

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All Grade 9 Mathematics short notes

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