Part 1 Module 2Set operations, Venn diagrams

Set operations

Let U = {x|x is an English-language film} Set A below contains the five best films according to

the American Film Institute. Set B below contains the five best films according to

TV Guide. Set C below contains the five most passionate films

according to the American Film Institute.

Set operations

A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia}

B= {Casablanca, The Godfather Part 2, The Wizard of Oz, Citizen Kane, To Kill A Mockingbird}

C = {Gone With the Wind, Casablanca, West Side Story, An Affair To Remember, Roman Holiday}.

Set operations

Form a new set whose elements are those that sets A and B have in common

A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia}

B= {Casablanca, The Godfather Part 2, The Wizard of Oz, Citizen Kane, To Kill A Mockingbird}

{Citizen Kane, Casablanca}This set is called the INTERSECTION of A and B, denoted

A ∩ B.

A ∩ B = {Citizen Kane, Casablanca}

Set operations

Find B ∩ C

B= {Casablanca, The Godfather Part 2, The Wizard of Oz, Citizen Kane, To Kill A Mockingbird}

C = {Gone With the Wind, Casablanca, West Side Story, An Affair To Remember, Roman Holiday}

B ∩ C = {Casablanca}

Set operations

A different operation: form a new set that contains all the elements of A along with all the elements of B.

A = {Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia}

B= {Casablanca, The Godfather Part 2, The Wizard of Oz, Citizen Kane, To Kill A Mockingbird}

{Citizen Kane, Casablanca, The Godfather, Gone With the Wind, Lawrence of Arabia, The Godfather Part 2, The Wizard of Oz, To Kill A Mockingbird}

This set is called the union of A with B, denoted A ∪ B.

Set operations

We have encountered three basic set operations (including something from Part 1 Module 1).

IntersectionS ∩ T = {x|x∈ S and x∈ T}

UnionS ∪ T = {x|x ∈ S or x ∈ T}.

ComplementS′ = {x|x ∉ S}.

Venn Diagrams

A Venn diagram is a drawing in which sets are represented by geometric figures such as circles and rectangles.

Venn diagrams can be used to illustrate the relationships between sets, and the effects of set operations.

Venn diagrams are also used in other areas of mathematics, such as counting, probability and logic.

Venn Diagrams - Intersection

Let S, T represent any sets in a universe U. The Venn diagram below illustrates the effect of intersection. The shaded region corresponds to S ∩ T.

Venn Diagrams - Union

Let S, T represent any sets in a universe U. The Venn diagram below illustrates the effect of union. The shaded region corresponds to

S ∪ T.

Let S, T represent any sets in a universe U. The Venn diagram below illustrates the effect of complement. The shaded region corresponds to S′ .

Venn diagrams - Complement

Exercise #1

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}S = {3, 5, 8, 11}T = {3, 4, 6, 7, 8, 10, 11}V = {2, 5, 6, 7, 8}W = {1, 3, 5, 6}

Find ( V′ ∩ S′ ) ∩ ( W ∪ T )

A. { }B. {4, 10, 11}C. {1, 4, 10}D. None of these

Solution #1

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}S = {3, 5, 8, 11} T = {3, 4, 6, 7, 8, 10, 11}V = {2, 5, 6, 7, 8} W = {1, 3, 5, 6}

Find ( V′ ∩ S′ ) ∩ ( W ∪ T )First, find V′ and S′ V′ = {1, 3, 4, 9, 10, 11} S′ = {1, 2, 4, 6, 7, 9, 10}So V′ ∩ S′ = {1, 4, 9, 10}Next, T = {3, 4, 6, 7, 8, 10, 11} and W = {1, 3, 5, 6}W ∪ T = {1, 3, 4, 5, 6, 7, 8, 10, 11}

Finally, ( V′ ∩ S′ ) ∩ ( W ∪ T ) = {1, 4, 10}

Exercise #2

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}T = {3, 4, 6, 7, 8, 10, 11}V = {2, 5, 6, 7, 8}

Find (V ∩ T′)′

A. {1, 3, 4, 6, 7, 8, 9, 10, 11}B. {3, 4, 10, 11}C. {1, 3, 4, 6, 8, 9, 10, 11}D. None of these

Solution #2

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}T = {3, 4, 6, 7, 8, 10, 11}V = {2, 5, 6, 7, 8}

Find (V ∩ T′)′

T′ = {1, 2, 5, 9} and V = {2, 5, 6, 7, 8}so V ∩ T′ = {2, 5}

Since V ∩ T′ = {2, 5} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}(V ∩ T′)′ = {1, 3, 4, 6, 7, 8, 9, 10, 11}

EXAMPLE 1.2.1 Venn diagrams

From your text: EXAMPLE 1.2.1 #13On the Venn diagram below, shade the region corresponding to

A ∪ B′ .

A B

U

Solution

To shade the region corresponding to A ∪ B′ , we need to understand the meanings of both terms, and understand the operation of union.

AIn the context of a shaded figure, A means that everything within set A is shaded.

B′In the context of a shaded figure, B′ means that everything outside set B is shaded.

Solution, page 2

Now, apply the inclusive operation of union to the shaded figures for A and B′The shaded figure for A ∪ B′ will include all the shading from the A figure along

with all the shading from the B′ figure.

AB′ΥΝΙΟΝA ∪ B′

Exercise #3

Select the Venn diagram whose shaded region corresponds to

A′ ∪ B′.

Solution #3

For A′ ∪ B′ ,we need to draw the shaded figure for A′, the shaded figure for B′, and then perform the union of the shading from those two figures.

A′Β′UNIONA′ ∪ B′

Exercise #4

Select the Venn diagram whose shaded region corresponds to

(A ∩ B)′.

Solution #4

To make the shaded figure (A ∩ B)′, we need to first make the shaded figure for A ∩ B, and then apply the idea of complement.

INTERSECTIONA ∩ ΒΧΟΜΠΛΕΜΕΝΤ(Α ∩ Β)′ΑΒ

DeMorgan’s Laws

In the previous two exercises we saw that the shaded figure for (A ∩ B)′ is identical to the shaded figure for A′ ∪ B′. This means that A′ ∪ B′ and (A ∩ B)′ are equivalent operations. This confirms one of the following general facts, which are known as DeMorgan’s Laws for Set Mathematics.

For any sets S, T( S ∪ T)′ = S′ ∩ T′( S ∩ T)′ = S′ ∪ T′“The complement of a union is the intersection of

the complements; the complement of an intersection is the union of the complements”

Exercise #5, DeMorgan’s Laws

Let U = {a, b, c, d, e, f, g}

T = {c, e}

V = {a, d, e}

Find ( T′ ∪ V′ )′A. {e}

B. {a, c, d, e}

C. {b, f, g}

D. None of these

Solution #5

U = {a, b, c, d, e, f, g} T = {c, e} V = {a, d, e}.Find ( T′ ∪ V′ )′

We will first simplify ( T′ ∪ V′ )′ by applying one of DeMorgan’s Laws, which states that we can distribute the outer complement onto both terms inside the parentheses, if we also change the union to intersection:

( T′ ∪ V′ )′ = T ∩ V

Now, finish the calculation by evaluating T ∩ V:

T = {c, e} and V = {a, d, e}, so T ∩ V = {e}

Alternative Solution #5

U = {a, b, c, d, e, f, g} T = {c, e} V = {a, d, e}Find ( T′ ∪ V′ )′

Instead of using one of DeMorgan’s Laws to simplify first, we can find the elements of T′, the elements of V′, perform the union of those two sets, and then the complement.

T′ = {a, b, d, f, g}V′ = {b, c, f, g}So, T′ ∪ V′ = {a, b, c, d, f, g} and finally ( T′ ∪ V′ )′ = {e}

Exercise #6

LIKE EXAMPLE 1.2.3 from your text:On the Venn diagram below, shade the region

corresponding to B′ ∩ (A ∪ C′)

A B

U

C

Solution #6

To shade the region corresponding to B′ ∩ (A ∪ C′), we must find the shaded figure for B′, the shaded figure for A ∪ C′, and the find the intersection of those shaded figures.

B′ µεανσ τηατ ωε σηαδε εϖερψτηινγ ουτσιδε οφ σετ Β.

Β′

Solution #6, page 2

The shaded figure for A ∪ C′ requires that we draw the shaded figure for A, and shaded figure for C′, and apply union to those two figures.

AC′UNIONA ∪ C′

Solution #6, page 3

Now that we have produced the shaded figure for B′ and the shaded figure for A ∪ C′, we intersect those figures to get the shading for B′ ∩ (A ∪ C′).

B′INTERSECTIONA ∪ C′B′ ∩ (A ∪ C′)

Exercise #7

Select the shaded figure for ( C ∩ B′ ) ∪ A′