Solve problems using set notation, operations, and Venn diagrams

Alberta Grade 12 Mathematics

Logical Reasoning

Representing groups, combining information, and checking totals

A survey can describe the same people in more than one way. Some students may play soccer, some may play volleyball, and some may play both. Set notation gives a concise way to describe these groups. A Venn diagram makes overlaps visible. In this lesson, you will translate between a problem’s wording, a diagram, and set notation. First identify the full group being studied. Then decide which region each count describes.

What you will learn

Read and write set notation for groups of people or objects.
Use union, intersection, difference, and complement to describe set relationships.
Fill and interpret two-set and three-set Venn diagrams.
Use given information to find unknown region counts and check the total.

1. Review: groups and totals

A set is a collection of clearly identified objects or people. For example, a set might contain every student who plays soccer. The objects in a set are called its elements. An element belongs to a set when it meets the condition used to define that set.

The universal set, written

U

, is the full group being considered. If a survey includes 40 students, those students make up

U

. A set names a group; a count tells how many elements are in that group. The notation

n(U)

means the number of elements in the universal set.

Before solving a problem, identify the universal set and read each count carefully. Ask whether a count describes a whole set, an overlap, a set-only region, or people outside the sets. This helps prevent counting anyone twice.

n(U)=number of elements in U

$U$ names the full group under consideration.
A Venn diagram region represents a specific group of elements.
In a completed diagram, each element belongs in exactly one disjoint region.

2. Set notation and operations

Sets are usually named with capital letters. Suppose

S

is the set of students who play soccer and

V

is the set of students who play volleyball. The notation x∈ S means that

x

belongs to

S

. The notation

x\notin S

means that

x

does not belong to

S

.

The intersection, written

S\cap V

, contains elements in both sets. The union, written

S\cup V

, contains elements in at least one of the sets. In this context, “or” means one set or the other or both. The overlap is included in the union.

The complement of

S

contains elements in the universal set that are not in

S

. It may be written S' or

S^c

. A complement always depends on the universal set. The difference

S\setminus V

means elements in

S

but not in

V

.

In a two-set Venn diagram, each circle represents a set and the overlap represents the intersection. The area inside either circle represents the union. The area in the rectangle but outside both circles represents elements in neither set.

To find a union count from two set totals, add the totals and subtract the intersection once. The overlap was included in each set total, so it was counted twice before the subtraction.

n(S\cup V)=n(S)+n(V)-n(S\cap V)

$S\cap V$ means in both sets.
$S\cup V$ means in at least one set.
S' means in the universal set but not in $S$ .
$S\setminus V$ means in $S$ but not in $V$ .

3. Filling and checking Venn diagrams

For two sets, fill the overlap first when its count is given. Then subtract that overlap from each set total to find the set-only regions. Finally, subtract all the inside regions from the universal total to find the neither region.

For three sets, the centre region represents elements in all three sets. A region where exactly two circles overlap represents elements in those two sets but not the third. If a pair count includes the centre, subtract the all-three count to find the exactly-two region.

A set total includes every region inside its circle. For example, the total in

S

includes elements in

S

only and elements in any overlaps involving

S

. It is not the same as the

S

-only region.

Once all regions are filled, add the disjoint regions. Their sum must equal the universal-set total. If it does not, check how you interpreted the wording and then check the arithmetic.

n(U)=sum of all disjoint regions

Place the most specific overlap information first.
Distinguish “in a set” from “in that set only.”
Use the universal total to find or check the outside region.

Worked example

Two school clubs

A group of 36 students is surveyed. Seventeen are in the art club, 14 are in the robotics club, and 6 are in both. Find how many are in at least one club and how many are in neither club. Let

A

be the art-club members and

R

the robotics-club members.

Club membership

Counts in each disjoint region of the 36-student group.

Place the overlap
The students in both clubs belong in the intersection region. This group is included in both club totals.
$n(A\cap R)=6$
Find the club-only regions
Subtract the overlap from each club total. That leaves students in each club but not the other.
$n(A\setminus R)=17-6=11,\quad n(R\setminus A)=14-6=8$
Find at least one and neither
Add the three inside regions to find the number in at least one club. Subtract that count from the full group to find the number in neither club.
$n(A\cup R)=11+6+8=25,\quad 36-25=11$

Answer: Twenty-five students are in at least one club, and 11 are in neither club.

Check: The four disjoint regions total

11+6+8+11=36

, matching the surveyed group. The union also agrees with

17+14-6=25

.

Worked example

Three preferred activities

In a group of 50 students, 26 like hiking, 23 like cycling, and 20 like camping. Twelve like hiking and cycling, 10 like hiking and camping, and 9 like cycling and camping. Each pair count includes students who like all three. Six students like all three. Find the number who like exactly one activity, at least one activity, and none.

Activity preferences

Each count is placed in exactly one region.

Find the exactly-two regions
Each pair count includes the six students who like all three activities. Subtract six from each pair count to find the students who like exactly that pair. For example, the hiking-and-cycling-only region has $12-6=6$ students. The hiking-and-camping-only region has $10-6=4$ , and the cycling-and-camping-only region has $9-6=3$ .
$12-6=6,\quad 10-6=4,\quad 9-6=3$
Find the one-activity regions
For each activity, subtract the other regions inside its circle from its total. For hiking, subtract the hiking-and-cycling-only region, the hiking-and-camping-only region, and the centre. Use the same reasoning for cycling and camping.
$26-6-4-6=10,\quad 23-6-3-6=8,\quad 20-4-3-6=7$
Find exactly one, at least one, and none
Exactly one is the sum of the three single-activity regions. At least one is the sum of every region inside the circles. Subtract that union from the group total to find how many like none.
$10+8+7=25,\quad 10+8+7+6+4+3+6=44,\quad 50-44=6$

Answer: Twenty-five students like exactly one activity, 44 like at least one activity, and 6 like none.

Check: The regions inside the circles total 44. Adding the 6 who like none gives the group total of 50.

Worked example

Reading a set description

A library has 42 books in a display. Let

F

be the set of fiction books and

N

the set of new books. There are 19 fiction books, 16 new books, and 7 books that are both fiction and new. How many books are fiction but not new? How many are neither fiction nor new? Write the group that is new or fiction using set notation.

Books in a display

Counts in each disjoint region of the 42-book display.

Find fiction but not new
The fiction total includes the books that are also new. Subtract the overlap to leave fiction books only.
$n(F\setminus N)=19-7=12$
Find the union
The union means new or fiction, including books that are both. Add the two set totals and subtract the overlap once.
$n(F\cup N)=19+16-7=28$
Find neither
The books outside both sets are the display total minus the union. This is the complement of the union within the display.
$42-28=14$

Answer: Twelve books are fiction but not new. Fourteen are neither. The group that is new or fiction is

F\cup N

.

Check: The disjoint regions are 12 fiction-only, 7 both, 9 new-only, and 14 neither. Their total is 42.

Common mistakes and how to avoid them

Adding two set totals to find the union without adjusting for the overlap.

Correction: Subtract the intersection once because its members were included in both set totals.

Treating a pair count in a three-set problem as the count in exactly those two sets.

Correction: If the pair count includes the all-three region, subtract that region to find the exactly-two count.

Assuming “or” means one set but not both.

Correction: Union means in at least one set, including the overlap.

Forgetting the members outside all circles.

Correction: Use the universal total to find or check the neither region.

Lesson summary

Define the universal set before interpreting counts.
Intersection means membership in both named sets; union means membership in at least one.
A complement is taken relative to the stated universal set.
Venn regions should be disjoint so each element is counted once.
Check that all region counts add to the universal-set total.

Check your understanding

Question 1

A group has 30 people. Twenty are in set

A

, 13 are in set

B

, and 8 are in both. How many are in

A\cup B

?

25
33
21
17

Show answer and explanation

25

The overlap is included in both set totals, so subtract it once:

20+13-8=25

.

Question 2

Which notation means “in

P

but not in

Q

”?

$P\cup Q$
$P\cap Q$
$P\setminus Q$
Q'

Show answer and explanation

P\setminus Q

P\setminus Q

names the members of

P

that are not members of

Q

.

Question 3

A group of 28 people has 19 in at least one of two sets. How many are in neither set?

9
19
47
10

Show answer and explanation

9

The neither region is outside the union, so subtract the union count from the universal total:

28-19=9

.

Key terms

Set: A collection of objects or people defined by a condition.
Universal set: The full group being considered in a problem, written $U$ .
Element: An object or person that belongs to a set.
Intersection: The set of elements that belong to all of the named sets.
Union: The set of elements that belong to at least one of the named sets.
Complement: The elements in the universal set that are not in the named set.
Venn diagram: A diagram that uses overlapping regions to represent sets and their relationships.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation LR2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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LR2 · Solve problems using set notation, operations, and Venn diagrams

Alberta Grade 12 Mathematics

Representing groups, combining information, and checking totals

What you will learn

1. Review: groups and totals

2. Set notation and operations

3. Filling and checking Venn diagrams

Two school clubs

Three preferred activities

Reading a set description

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review