Set Solutions and Venn Diagram.pps

7/26/2019 Set Solutions and Venn Diagram.pps

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Part 1 Module 2Set operations, Venn diagrams


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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.


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Set operations

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

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

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


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Set operations

Form a new set whose elements are those that sets A and Bhave 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 INTERSECTIONof A and B, denoted A

B.

AB={Citizen Kane, Casablanca}


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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, AnAffair To Remember, Roman Holiday}

BC= {Casablanca}


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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 KillA Mockingbird}

This set is called the unionof A with B, denoted A B.


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Set operations

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

IntersectionS T = {x|xS and xT}

Union

S

T = {x|x

S or x

T}.

Complement

S= {x|x S}.


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Venn Diagrams

AVenn diagramis a drawing in which sets arerepresented by geometric figures such as circles andrectangles.

Venn diagrams can be used to illustrate therelationships between sets, and the effects of setoperations.

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


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Venn Diagrams - Intersection

Let S, T represent any sets in a universe U. The Venndiagram below illustrates the effect of

intersection. The shaded region corresponds toS T.


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Venn Diagrams - Union

Let S, T represent any sets in a universe U. The Venndiagram below illustrates the effect of

union. The shaded region corresponds to

S

T.


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Let S, T represent any sets in a universe U. The Venn

diagram below illustrates the effect of

complement.The shaded region corresponds toS.

Venn diagrams - Complement


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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 ( VS) ( WT )

A. { }B. {4, 10, 11}

C. {1, 4, 10}

D. None of these


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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 ( VS) ( WT )

First, find V and S

V= {1, 3, 4, 9, 10, 11} S= {1, 2, 4, 6, 7, 9, 10}

So VS= {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, ( VS) ( WT ) = {1, 4, 10}


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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


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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}

Sin! V T= {2, 5} and " = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

(V T)= {1, 3, 4, 6, 7, 8, 9, 10, 11}


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EXAMPLE 1.2.1Venn diagrams

From your text:

EXAMPLE 1.2.1 #13

#n t$! V!nn dia%ra& '!o, s$ad! t$! r!%ion orr!s*ondin% to

A B.

A B

U


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Solution

To s$ad! t$! r!%ion orr!s*ondin% to A B, we need to understand the meaningsof both terms, and understand the operation of union.

shaded.

is shaded.


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Solution, page 2

+o, a** t$! in-si.! o*!ration of -nion to t$! s$ad!d fi%-r!s for A andB

The shaded figure for A Bwill include all the shading from the A figure alongwith all the shading from the Bfigure/

A BAB


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Exercise #3

S!!t t$! V!nn dia%ra& $os! s$ad!d r!%ion orr!s*onds to

AB.


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Solution #3

For AB,! n!!d to dra t$! s$ad!d fi%-r! for A, the shadedfigure for B, and then perform the union of the shading from those

two figures.AUNION

A

B


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Exercise #4

S!!t t$! V!nn dia%ra& $os! s$ad!d r!%ion orr!s*onds to

(AB).


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Solution #4

To make the shaded figure (AB), we need to first make theshaded figure for AB, and then apply the idea of

complement.

INTEA )


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DeMorgans Laws

In the previous two exercises we saw that the shaded figure for (AB)isidentical to the shaded figure for AB. This means that AB and(AB) are equivalent operations. This confirms one of the followinggeneral facts, which are known as DeMorgans Laws for SetMathematics.

For any sets S, T

( S T)= ST

( S T)= ST

The compeme!t o" u!$o! $% the $!ter%ect$o! o" thecompeme!t%& the compeme!t o" ! $!ter%ect$o! $% theu!$o! o" the compeme!t%'


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Exercise #5, DeMorgans Laws

!t" = {a, ', , d, !, f, %}

T = {, !}

V = {a, d, !}

Find ( T)

/ {!}

/ {a, , d, !}

/ {', f, %}

/ +on! of t$!s!


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Solution #5

" = {a, ', , d, !, f, %}

T = {, !}

V = {a, d, !}/

Find ( TV)

W! i first si&*if ( TV)' a**in% on! of !or%ans as, $i$stat!s t$at ! an distri'-t! t$! o-t!r o&*!&!nt onto 'ot$ t!r&s insid! t$!*ar!nt$!s!s, if ! aso $an%! t$! -nion to int!rs!tion

( TV)= T V

+o, finis$ t$! a-ation ' !.a-atin% T V

T = {, !} and V = {a, d, !}, so T V = {!}


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Alternative Solution #5

" = {a, ', , d, !, f, %} T = {, !} V = {a, d, !}

Find ( TV)

nst!ad of -sin% on! of !or%ans as to si&*if first, ! an find t$! !!&!ntsof T, t$! !!&!nts of V, *!rfor& t$! -nion of t$os! to s!ts, and t$!n t$!

o&*!&!nt/

T= {a, ', d, f, %}

V= {', , f, %}

So, TV= {a, ', , d, f, %}

and fina( TV)= {!}


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Exercise #6

: :;


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Solution #6

To s$ad! t$! r!%ion orr!s*ondin% to B(A C), we mustfind the shaded figure for B, the shaded figure for A C,and the find the intersection of those shaded figures.

B


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Solution #6, page 2

T$!shaded figure for A Crequires that we draw the shaded figure for A,and shaded figure for C, and apply union to those two figures.

ACACUNION


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Solution #6, page 3

+o t$at ! $a.! *rod-!d t$! s$ad!d fi%-r! for Band the shaded figure for A C,we intersect those figures to get the shading for B(A C).

B(A C)BINTERSECTIONA C


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Exercise #7

Select the shaded figure for ( )

Set Solutions and Venn Diagram.pps

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