Section 2.3Using Venn Diagrams to Study

Set Operations

Math in Our World

Learning Objectives

Illustrate set statements involving two sets with Venn diagrams.

Illustrate set statements involving three sets with Venn diagrams.

Use De Morgan’s laws.Use Venn diagrams to decide if two sets

are equal.Use the formula for the cardinality of a

union.

Illustrating a Set Statement with a Venn Diagram

Step 1 Draw a diagram for the sets, with Roman numerals in each region.

Step 2 Using the Roman numerals, list the regions for each set.

Step 3 Find the set of numerals that correspond to the set given in the set statement.

Step 4 Shade the area corresponding to the set of numerals found in step 3.

Region I represents the elements in set A that are not in set B.

Region II represents the elements in both sets A and B.Region III represents the elements in set B that are not

in set A.Region IV represents the elements in the universal set

that are in neither set A nor set B.

A

IIIIII

U B

IV

EXAMPLE 1 Drawing a Venn Diagram

Draw a Venn diagram to illustrate the set (A B).

SOLUTIONStep 1 Draw the diagram and label each area.

Step 2 From the diagram, list the regions that make up each set.

U = {I, II, II, IV} A = {I, II} B = {II, III}

Step 3 Using the sets in step 2, find (A B).First, all of I, II, and III are in either A or B, so A B = {I, II, II}.

The complement is (A B) = {IV}.

Step 4 Shade region IV to illustrate (A B).

IIIIII

B

IV

AU

EXAMPLE 2 Drawing a Venn Diagram

Draw a Venn diagram to illustrate the set A B.

SOLUTIONStep 1 Draw the diagram and label each area. (same as Example 1)

Step 2 From the diagram, list the regions that make up each set.

U = {I, II, II, IV} A = {I, II} B = {II, III}

Step 3 Using the sets in step 2, find A B.First, B = {I, IV}. Of these two regions, I is also in set A, so

A B = {I}.

Step 4 Shade region I to illustrate A B.

IV

U B

IIIIII

A

Venn Diagrams with Three SetsRegion I represents the elements in set A but not in set B or set C.

Region II represents the elements in set A and set B but not in set C.

Region III represents the elements in set B but not in set A or set C.

Region IV represents the elements in sets A and C but not in set B.

Region V represents the elements in sets A, B, and C.

Region VI represents the elements in sets B and C but not in set A.Region VII represents the elements in set C but not in set A or set B.Region VIII represents the elements in the universal set U, but not in set A, B, or C.

IIIIA

II

IVVVI

VIIC

B

VIII

I II

IV

A

C

B

EXAMPLE 3 Drawing a Venn Diagramwith Three Sets

Draw a Venn diagram to illustrate the set A (B C).

SOLUTIONStep 1 Draw the diagram and label each area.

Step 2 From the diagram, list the regions that make up each set.

U = {I, II, III, IV, V, VI, VII, VIII} A = {I, II, IV, V}

B = {II, III, V, VI}C = {IV, V, VI, VII}

Step 3 Using the sets in step 2, first B C = {V, VI}, so (B C) = {I, II, III, IV, VII, VIII}.Of these regions, I, II, IV is also in set A, so A (B C) = {I, II, IV}.

Step 4 Shade regions I, II, IV to illustrate A (B C) .

IIII II

IVVVI

VIIVIII

De Morgan’s Laws

For any two sets A and B,

(A B) = A B(A B) = A B

The first law states that the complement of the union of two sets will always be equal to the intersection of the complements of each set.

The second law states that the complement of the intersection of two sets will equal the union of the complements of the sets.

EXAMPLE 4Using De Morgan’s Laws

If U = {a, b, c, d, e, f, g, h}, A = {a, c, e, g}, and B = {b, c, d, e}, find (A B) and A B.

SOLUTION

A B = {a, b, c, d, e, g}, so (A B) = {f, h}

A = {b, d, f, h} and B = {a, f, g, h},

so A B = {f, h}

Notice that (A B) and A B are equal, illustrating the first of De Morgan’s laws.

EXAMPLE 5Understanding Subset Notation

If U = {10, 11, 12, 13, 14, 15, 16}, A = {10, 11, 12, 13}, and B = {12, 13, 14, 15}, find (A B) and A B.

SOLUTION

A B = {12, 13}, so (A B) = {10, 11, 14, 15, 16}

A = {14, 15, 16} and B = {10, 11, 16}, so

A B = {10, 11, 14, 15, 16}

Notice that (A B) and A B are equal, illustrating the second of De Morgan’s laws.

EXAMPLE 6 Using a Venn Diagram to Show Equality of Sets

Use Venn diagrams to show that (A B) = A B.SOLUTIONStart by drawing the Venn diagram for (A B).This was done in Example 1, giving a result of…Next draw the Venn diagram for A B.Step 1 Draw a second Venn diagram with two sets.

Step 2 From the diagram, list the regions that make up each set.U = {I, II, II, IV} A = {I, II} B = {II, III}

Step 3 A = {III, IV} and B = {I, IV}, so

A B = {IV}.

Step 4 Shade region IV to illustrate A B.

IIIIII

B

IV

AU

IIIIII

B

IV

AU

EXAMPLE 7 Using Venn Diagrams to Decide If Two Sets Are Equal

Determine if the two sets are equal by using Venn diagrams:

(A B) C and (A C) (B C)

IVVVI

EXAMPLE 7 Using Venn Diagrams to Decide If Two Sets Are Equal

SOLUTION

Start with (A B) C.

The set A B = {I, II, III, IV, V, VI}.

Of these, IV, V, and VI are also in C, so (A B) C = {IV, V, and VI}.

Now let’s examine (A C) (B C).

The set A C = {IV, V},

and the set B C = {V, VI}.

(A C) (B C) = {IV, V, VI}.

Since the shaded areas are the same, the two sets are equal.

A

C

B

IVVVI

A

C

B

IIII II

IVVVI

VII VIII

IIII II

IVVVI

VII VIII

Cardinality of a Union

If n(A) represents the cardinal number of set A, then for any two finite sets A and B,

n(A B) = n(A) + n(B) – n(A B).

In words, the formula says that to find the number of elements in the union of A and B, you add the number of elements in A and B and then subtract the number of elements in the intersection of A and B.

EXAMPLE 8 Using the Formula for Cardinality of a Union

In a survey of 100 randomly selected freshmen walking across campus, it turns out that 42 are taking a math class, 51 are taking an English class, and 12 are taking both. How many students are taking either a math class or an English class?

SOLUTIONIf we call the set of students taking a math class A and the set of students taking an English class B, we’re asked to find n(A B). We’re told that n(A) = 42, n(B) = 51, and n(A B) = 12. So…

n(A B) = n(A) + n(B) – n(A B) = 42 + 51 – 12 = 81