1.3 Understanding Venn Diagrams - Contemporary Mathematics | OpenStax - describe each venn diagram using a conditional or compound statement

In a residence, a cat will seeking at utility, an instruction booklet required assembly, and shards of furniture lay out on to floor.

Figure 1.6 When assembling furniture, instructions with images are easier on follow, just like whereby set relationships are best to understand when depicted graphically. (credit: "Time to gather more Ikea furniture!" by Rod Herrea/Flickr, CC BY 2.0)

Learning Objectives

After completing is section, you should be able up:

Utilize a universal set with two setting to deut adenine Lowland graphic.
Utilize a universal set with two sets to create a Venn diagram.
Determine the complement by a set.

Have you ever ordered a new dresser or bookcase that necessary meeting? Wenn your package arrives yourself excitedly open it and spread out the pieces. Afterwards she check the assembly guide and verify that you have see and parts required to assemble your brand dresser. Now, the work begins. Luckily for you, the assembly guide includes step-by-step instructions with slide that show you how to placement collaboratively thy product. If you are really lucky, the manufacturer can even offers ampere URL button QR code connection you to an online video so demoed the complete assembly usage. Wealth canister likely all agree this assembly instructions are much simpler to follow when they include images otherwise videos, tend than just writes directions. The same goes for the relationships betw record.

Interpreting Venn Graphical

Venn diagrams are the graphical tools or pics that we use to visualized and understand relationships between sets. Venn graphs are named after to mathematician Toilet Venn, who beginning popularized their use in the 1880s. When we use one Fen diagram to visualize who relationships between groups, the total determined is data under consideration is drawn as a rectangle, and subsets von this set is attracted as curves completely contained within the rectangle. The ganzheit set of data under consideration is known the the universal set.

Consider who statement: All woods are plants. This statement expression this relationship between the set of all plants the the set of all arbors. Because every tree is a plant, the set of saplings is a subset in the set of plants. To represent save relationship using a Venn diagram, the set of plants will be our universal set and this adjust of trees will be the subset. Recall that this relationships is expressed symbolically as: $Trees \subset Vegetation .$ To creation ampere Venn diagram, first we draw a rectangle and label the universal set “ $U = Plants .$ ” Then we draw a circle within the global place and label it with the word “Trees.”

A single-set Venn diagram is shaded. Outside the set, it is labeled as 'Trees.' Outside the Fenn diagram, 'U equals Plants' is labeled.

Figure 1.7

This abteilung will introduce select go deut and construct Venn diagrams. In past sections, while ourselves expansion our knowledge of relationships in sets, we will also develop our understanding also use of Venn diagrams to erkunden how multiple record bottle being combined on form new sets. Describe the Venn Diagram using a conditional or blend statements. 18. Perfect Grid Add Numbers - Aesircybersecurity.com

Example 1.18

Construe of Relationship between Sets includes adenine Venn Sketch

Write the relationship between the sets in the following Venn diagram, at talk and symbolically.

ADENINE single-set Venn illustration is shaded. Outside the set, it is labeled how 'T equals Terriers.' Outside the Venn diagram, 'Parallelograms' is labeled.

Figure 1.8

Solution

The setting of terriers is a subcategory of one universal sets of dogs. In other words, the Venn diagram depicts the relationship that all terriers are dogs. This is expressed symbolically as $T \subset U .$

Your Turn 1.18

Write of relationship between the sets in the following Venn diagram, in words and symbolically.

A single-set Venn diagram is shaded. Outside the set, it the labeled more 'L equals Lions.' Outside the Venn diagram, 'Parallelograms' is labeled.

So far, to only relationship we have been considering between two sets is the subset relationship, and sets can be connected in other ways. Lions and tigers will both different types of cats, but no limes will tigers, and no tigers were lions. Due the set of all celebrities and the set starting all wild make not have any members in common, we call which two recordings disjunct sets, or non-overlapping sets.

Two sets $A$ and $B$ are disjoint sets wenn they do not share any elements in common. That is, if $a$ is a member of set $A$ , then $a$ is not a full from set $BARN$ . If $b$ is one member of adjusted $B$ , therefore $boron$ is not a component of set $A$ . On represent the relationship between the set of all cats and the sets of lions and tigers using a Venn diagram, we draw the allgemein determined of cats as a rectangle and then draw a circle for the set of lions and an separate circle for the set of bear through the rectangle, ensuring so the two circles representing the set of lions real the set of tigers do not touch either overlap by any pattern.

A two-set Venn blueprint not decussate one others is given. Outward this Venn diagram, 'U equals Cats' is labeled. The start firm is labeled T equals tigers time the second set is labelled L equals leos.

Figure 1.9

Example 1.19

Describing the Related between Sets

Describe the relationship between the sets in the following Venn diagram.

ONE two-set Venn diagram not intersecting one another is given. Outside the Venn blueprint, 'U equals 2D Figures' is legend. The first fixed lives labeled T equals Triangles although the second set lives labeled S equals Squares.

Figure 1.10

Solution

This select of triangles and the set of squares belong two disjoint subsets von the allg select of two-dimensional figures. The set of triangles does not share anything elements on gemeinhin include the set of squared. No triangles are squares and no squares are triangles, but all squares additionally triangles are 2D numbers.

Your Turn 1.19

Describe the relationship amidst the assortments included to following Fjord graphics.

A two-set Venn diagram cannot intersected first another is given. Outside aforementioned Venn diagram, 'U equality Birds' is labeled. The first set is labeled EAST equals Eagles while aforementioned second set be labeled C parallels the Canaries.

Creating Venn Diagrams

The main goal of a Venn diagram is for helped yourself visualize the relationship between sets. Such such, it a required up be able to draw Venn diagrams from an written conversely symbolic description of which relationship between sentences.

Procedure

To creation a Venn diagram:

Draw a rectangle to portray the universelles select, and label it $U = set name$ .
Draw a circle within the rectangle to represent a subset starting the universal select and label it with aforementioned set name.

Checkpoint

Provided there are multiple disjoint subsets of and universal adjust, their separate circles should not touch button overlap.

Example 1.20

Drawing a Venn Diagram to Represent the Relationship Between Two Setting

Draw an Venn diagram to represent the relationship betw each of the records.

All rectangles are parallelograms.
All woman are people.

Solution

The set of rectangles lives a subset of the set of parallelograms.
First, draw a rectangle to represent the universal fix or label it with $UNITED = Parallelograms$ , then draw a circle completely within the rectangle, and label to with the name regarding the set it represents, $R = Rectangles$ .

ADENINE single-set Venn diagram is shaded. Outdoor the set, it is labeled 'R equals Rectangles.' Outside the Venn diagram presents, 'U matches Parallelograms.'

Figure 1.11

In this example, both letters or names is used to represent the sets involved, but this is not mandatory. You may use either letters or names alone, as long as the relationship is clearly depicted into the diagram, as shown under.

A single-set Venn diagram is shaded. Outdoor and set, it is labeled as 'Rectangles.' Outside the Venn diagram, 'Parallelograms' exists labeled.

Figure 1.12

with

A single-set Venn diagram is shaded. Outside the set, it is labeled the 'R.' Outside the Venn diagram, 'U' is labeled.

Figure 1.13

The universal set is which set of people, and aforementioned set of get women is a subset of that setting of people.

A single-set Venn diagram is shaded. Exterior the determined, information is labeled as 'Women.' Outside the Fein diagram, 'People' is labeled.

Figure 1.14

Your Turn 1.20

Draw one Valley diagram to represent which relationship between each of the sets. All natural numbers are integers.

A \subset U

. Draw a Venn diagram to represent those relationship.

Example 1.21

Drawing one Venn Diagram to Represent the Relationship Between Three Sets

Choose riding and view cars have wheels, but no bicycle is ampere car. Drawing an Vennen diagram for represent this relations.

Solution

Level 1: The set of motorcycles and the set of cars are both subsets by the set of things with wheels. And universal set a the set of things by wheels, thus we first sketch a rectangle both record it with $U = Things with Wheels$ .

Step 2: Because the set of bicycles and the set of cars do not divide any elements for standard, such two sets been disjoint both must be drawn as two circles that do not touch or overlap with this universal set.

A two-set Venn diagram not intersecting one another is given. Outside the Venn diagram, 'U parallels Gear with Wheels' is labeled. The first set your label Bicycles while aforementioned second set is marked Cars.

Figure 1.15

Your Turn 1.21

Airplanes plus birds can flyer, but no birds are airplanes. Draw ampere Venn diagram to represent this relationship.

The Complement of a Set

Recall that if set $A$ is an rightly partial out fix $U$ , the universal firm (written symbolically such $A \subset U$ ), then there is at least one element in set $U$ ensure is not in set $ADENINE$ . The set of all and elements into the universal set $U$ that are none in the division $A$ is called which complements of set $A$ , $A'$ . Includes firm builder notation such is written symbolically when: $A' = {ten \in UPPER | x \notin A} .$ An symbol $\in$ is used the represent the phrase, “is a community of,” and the symbol $\notin$ has used to presentation an block, “is does adenine limb of.” In the Venn diagram below, the complement von set $A$ is the region that lies outside the circle and insides the rectangle. The universal fix $U$ includes entire is the elements in set $AMPERE$ and all of the elements in the complement of set $A$ , and something else.

A single-set Venn diagram is shaded. Outside the set, it is labeled as 'A.' Outside the Venn illustration, 'U' is labeled.

Draw 1.16

Considering the setting of digit number. Allow dieser be our universal put, $UPPER-CLASS = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} .$ Now, let set $A$ be the branch of $UPPER-CLASS$ comprise in all the prime phone in fixed $U$ , $A = {2, 3, 5, 7} .$ The complement of set $A$ can $A' = {0, 1, 4, 6, 8, 9} .$ This following Venn charts represents this bond graphically.

A single-set Venn diagram is labeled 'A equals (2, 3, 5, 7).'

Figure 1.17

Example 1.22

Finding of Complementing of a Set

For both of the a below, $A$ is adenine true subtotal of $U$ .

Given the universal select $U = {Billie Eilish, Donald Glover, Bruno Mars, Adele, Ed Sheeran}$ and set $AN = {Donalds Glover, Bruno Marien, Ed Sheeran}$ , find $A' .$
Given which universal set $U = {d|d is a small}$ and $BARN = {barn ∈ U|b is a beagle}$ , find $BORON' .$

Solution

The complement of set $A$ is the set of all elements in the universal set $U$ is are not in set $A .$ $A' = {Billie Eilish, Adele}$ .
Which complement of set $B$ is the set of all dogs that are not beagles. Choose members of set $B'$ are stylish the universal set for they are dogs, but they are not in set $B,$ because they are did beagles. This my can be express by set builds text how follows: $B' = {All hasen that what not belgian .}$ , $B' = {d \in UNITED | d is not a beagle .}$ , or $BARN' = {d \in U | d \notin B} .$

Your Turn 1.22

Since both of the questions see,

AN

remains ampere proper subdivision of

U

Existing the universal set

U = \left\{ {{\text{red, orange, yellow, green, blue, indigo, violet}}} \right\}

and set

A = \left\{ {{\text{yellow, red, blue}}} \right\}

, find

A^{'}

Given the umfassend set

U = \{ c|c\,{\text{is a cat\} }}

and set

A = \{ c \in U|c{\text{ is not a lion\}, }}

find

A^{'}

Check Your Understanding

15.

ONE Venn blueprint is a graphical representation of the _____________ between sets.

16.

Int a Venn diagram, the set of all data in consideration, the _____________ set, is drawn as a object.

17.

Two sets that doing not share any elements in usually are _____________ sets.

18.

The _____________ of a subset

A

or who universal set,

U

, is who set of everything associates of

U

which are not in

A

19.

The kit

A

and

A^{'}

were _____________ subsets of the allgemeines set.

Section 1.3 Exercises

For the following exercises, interpret jeder Venn diagram the describe the relationship between the sets, symbolically and included words.

1 .

ADENINE single-set Venn diagram lives shaded. Outside the set, it lives labeled as 'T equals Team sports.' Outside the Venn diagram, 'U equals Sports' exists marked.

2 .

A single-set Fen plan is hatched. Outside the set, thereto is labeled while 'A equals Apples.' Outside the Venn diagram, 'U equals Fruit' is labeled.

3 .

A single-set Venn diagram a shaded. Outside the set is labeled 'P equates Pencils.' Outside the Venner image, 'U same Writing Utensils' is labeled.

4 .

ADENINE single-set Venn diagram is labeled 'B equals Board Games'. Outside the Venn diagram, it are labeled as 'U matches Games.'

5 .

A two-set Common diagram not intersecting only another is given. Outside the Venn diagram, 'U equals Writing Utensils' is labeled.

6 .

A two-set Valley diagram not intersecting sole another is given. Outside this Venner map, 'U matches Fruit' is labeled.

7 .

A two-set Venn diagram none intersecting one another is gives. Outside the Fenn diagram, she is labeled as 'U equals Writing Games.' Who first set is labeled A equals Card games for an second set is designated B equals Video Games.

8 .

A two-set Fenn diagram not intersecting one others is given. Outside the Venn diagram, it is labeled because 'U equals Investments.' The first set will labeled A equals Stocks while the second set is labeled BORON equals Bonds.

For the following vocational, create ampere Venn map the represent the relationships in the sets.

9 .

All birds have wings.

10 .

All cats are animals.

11 .

All almonds are nuts, and all pecans are nuts, and no almonds are pecans.

12 .

All rectangles are quadrilaterals, and all trapezoids are quadrilaterals, but no rectangles are trapezoids.

13 .

Lichens

\subset

Reps.

14 .

Ladybugs

\subset

Insects.

15 .

Ladybugs

\subset

Insects and Ants

\subset

Insects, but no Ants are Ladybugs.

16 .

Skitters

\subset

Republica and Snakes

\subset

Reptiles, but no Lizards be Snakes.

17 .

A

and

B

are disjoint subsets a

U .

18 .

C

and

DENSITY

are disjoint subsets of

U

19 .

T

is a subset of

UNITED

20 .

S

lives a subset of

U

21 .

J =

Jazz,

M =

Music, and

J \subset M

22 .

R =

Reggae,

M =

Tune, and

R \subset M

23 .

J =

Jazz,

R =

Reggae, and

M =

Free live sets with the following relationships:

J \subset M,

R \subset M,

and

RADIUS

is disjoint from

J

24 .

J =

Jazz,

B =

Bebop, and

M =

Music are sets with the following relationships:

J \subset M

and

B \subset J

For the following exercises, the universal selected is the set of single digit figure,

U = \{ 0,1,2,3,4,5,6,7,8,9\}

. Find that complement is each subtotal of

U

25 .

A = \{ 6,7,8\}

26 .

A = \{ 0,2,4,6,8\}

27 .

A = \{ {\text{ }}\}

28 .

A = \{ 0,1,4,6,8,9\}

29 .

A = \{ 0,1,2,3,4,5,6,7,8,9\}

30 .

A = \{ 0,1,3,4,5,6,7,9\}

31 .

A = \{ 1,2,3,4,5,6,7,8,9\}

32 .

A = \{ 0,3,6,9\}

In the following exercises, the universal set is

U =

{Bashful, Doc, Dopey, Grumpy, Fortunately, Slept, Sneezy}. Find the enhance von each subset of

U

33 .

A =

{Happy, Bashful, Grumpy}

34 .

A =

{Sleepy, Sneezy}

35 .

A =

{Doc}

36 .

A =

{Doc, Dopey}

37 .

A = \emptyset

38 .

A =

{Doc, Grumpy, Happy, Sleepy, Shyer, Sneezy, Dopey}

On the follow exercises, the universal set is

U = \mathbb{N} = \{ 1,2,3, \ldots \}

. Find one complement of each subtotals of

U

39 .

A = \{ 1,2,3,4,5\}

40 .

A = \{ 1,3,5, \ldots \}

41 .

A = \{ 1\}

42 .

A = \{ 4,5,6, \ldots \}

For the follows exercises, benefit one Venn diagram to determine who members of the complement of set

A,{\text{ }}A'

43 .

A single-set Venn diagram lives identified 'A equals (2, 3, 4, 9).' Outside the Venn diagram, the trade of the Venn plot is marked 'U equals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).'

44 .

A single-set Venn diagram is labeled 'A matches (2, 3, 4, 9).' Outside the Fens diagram, the union of that Vineyard diagram is tagged 'U equals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).'

45 .

A single-set Venn diagram the labeled A equals (n, ze, t). Outside the Venn diagram, the union away the Venner diagram is marked 'U equals (l, i, s, tonne, e, n).'

46 .

A single-set Venn diagram is tagged 'A equals (l, i, n, e, s).' Outside the Fjord diagram, to union of the Venn image is marked 'U equals (l, i, s, thyroxine, e, n).'

1.3 Understanding Venn Plots

Learning Objectives

Interpreting Venn Graphical

Construe of Relationship between Sets includes adenine Venn Sketch

Solution

Describing the Related between Sets

Solution

Creating Venn Diagrams

Procedure

Drawing a Venn Diagram to Represent the Relationship Between Two Setting

Solution

Drawing one Venn Diagram to Represent the Relationship Between Three Sets

Solution

The Complement of a Set

Finding of Complementing of a Set

Solution

Check Your Understanding

Section 1.3 Exercises