How To Find Elements In A Set

Information technology is natural for the states to classify items into groups, or sets, and consider how those sets overlap with each other. We tin use these sets understand relationships between groups, and to clarify survey data.

Nuts

An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Whatever collection of items can form a prepare.

Set

A ready is a drove of distinct objects, chosen elements of the set

A fix can be defined by describing the contents, or by list the elements of the set up, enclosed in curly brackets.

Instance 1

Some examples of sets defined by describing the contents:

1. The prepare of all even numbers
2. The prepare of all books written about travel to Chile

Answers

Some examples of sets defined by listing the elements of the set:

1. {one, 3, 9, 12}
2. {red, orange, xanthous, green, blue, indigo, purple}

A set simply specifies the contents; order is not important. The set represented by {i, two, 3} is equivalent to the set {3, 1, 2}.

Annotation

Commonly, we volition use a variable to correspond a set, to brand it easier to refer to that gear up afterwards.

The symbol ∈ means "is an chemical element of".

A set that contains no elements, { }, is called the empty set and is notated ∅

Case 2

Let A = {1, two, 3, 4}

To notate that 2 is element of the set, we'd write 2 ∈ A

Sometimes a collection might non comprise all the elements of a set. For example, Chris owns iii Madonna albums. While Chris's collection is a set, we can also say information technology is a subset of the larger set of all Madonna albums.

Subset

A subset of a set up A is another fix that contains but elements from the set A, just may not incorporate all the elements of A.

If B is a subset of A, nosotros write B ⊆ A

A proper subset is a subset that is non identical to the original set—it contains fewer elements.

If B is a proper subset of A, we write B ⊂ A

Instance 3

Consider these iii sets:

A = the set of all even numbers
B = {2, 4, vi}
C = {2, 3, 4, half-dozen}

Here B ⊂ A since every chemical element of B is also an even number, so is an element of A.

More formally, we could say B ⊂ A since if x ∈B, then x ∈ A.

It is likewise true that B ⊂ C.

C is non a subset of A, since C contains an element, 3, that is not independent in A

Example iv

Suppose a fix contains the plays "Much Ado About Zip," "MacBeth," and "A Midsummer's Night Dream." What is a larger set this might be a subset of?

In that location are many possible answers here. 1 would exist the set up of plays by Shakespeare. This is too a subset of the set of all plays e'er written. Information technology is also a subset of all British literature.

Effort Information technology Now

The set A = {1, iii, v}. What is a larger set this might exist a subset of?

Union, Intersection, and Complement

Usually sets interact. For example, you and a new roommate make up one's mind to take a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.

Union, Intersection, and Complement

The matrimony of two sets contains all the elements contained in either set (or both sets). The matrimony is notated A ⋃ B.More formally, ten ∊ A ⋃ B if ten ∈ A or x ∈ B (or both)

The intersection of two sets contains but the elements that are in both sets. The intersection is notated A ⋂ B.More formally, 10 ∈ A ⋂ B if x ∈ A and x ∈ B.

The complement of a ready A contains everything that is not in the prepare A. The complement is notated A', or A^c, or sometimes ~A.

Example 5

Consider the sets:

A = {cherry, green, blueish}
B = {red, yellow, orangish}
C = {carmine, orange, yellowish, green, blue, regal}

Find the post-obit:

1. Observe A ⋃ B
2. Discover A ⋂ B
3. Find A^c ⋂ C

Answers

1. The union contains all the elements in either set: A ⋃ B = {crimson, green, blue, yellow, orange} Notice nosotros only list red once.
2. The intersection contains all the elements in both sets: A ⋂ B = {ruddy}
3. Hither we're looking for all the elements that are not in fix A and are also in C. A^c ⋂ C = {orangish, yellowish, purple}

Try It Now

Using the sets from the previous example, find A ⋃ C and B ^c ⋂ A

Notice that in the instance higher up, information technology would be hard to but inquire for A^c , since everything from the color fuchsia to puppies and peanut butter are included in the complement of the fix. For this reason, complements are usually only used with intersections, or when nosotros accept a universal gear up in place.

Universal Set

A universal set is a fix that contains all the elements we are interested in. This would have to be defined by the context.

A complement is relative to the universal set, and thenA^c contains all the elements in the universal set that are not in A.

Instance 6

1. If we were discussing searching for books, the universal set might be all the books in the library.
2. If we were group your Facebook friends, the universal set would be all your Facebook friends.
3. If you were working with sets of numbers, the universal prepare might be all whole numbers, all integers, or all real numbers

Example vii

Suppose the universal set is U = all whole numbers from 1 to 9. If A = {1, 2, 4}, then A^c = {three, 5, half dozen, 7, 8, nine}.

As we saw earlier with the expressionA^c ⋂ C, set operations can exist grouped together. Group symbols can exist used like they are with arithmetic – to force an order of operations.

Example eight

Suppose H = {cat, dog, rabbit, mouse}, F = {domestic dog, moo-cow, duck, pig, rabbit}, andWestward = {duck, rabbit, deer, frog, mouse}

1. Find (H ⋂ F) ⋃ Due west
2. Find H ⋂ (F ⋃ West)
3. Find (H ⋂ F) ^c ⋂ W

Solutions

1. We start with the intersection: H ⋂ F = {canis familiaris, rabbit}. Now we marriage that result with W: (H ⋂ F) ⋃ W = {canis familiaris, duck, rabbit, deer, frog, mouse}
2. Nosotros start with the union: F ⋃ W = {dog, cow, rabbit, duck, pig, deer, frog, mouse}. Now we intersect that result with H: H ⋂ (F ⋃ W) = {canis familiaris, rabbit, mouse}
3. We commencement with the intersection: H ⋂ F = {dog, rabbit}. At present we want to find the elements of Westward that are not in H ⋂ F.(H ⋂ F) ^c  ⋂ W = {duck, deer, frog, mouse}

Venn Diagrams

To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, edifice on a like thought used by Leonhard Euler in the eighteenth century. These illustrations now called Venn Diagrams.

Venn Diagram

A Venn diagram represents each ready past a circumvolve, usually drawn inside of a containing box representing the universal ready. Overlapping areas point elements common to both sets.

Basic Venn diagrams tin can illustrate the interaction of 2 or three sets.

Instance 9

Create Venn diagrams to illustrate A ⋃ B, A ⋂ B, and A ^c ⋂ B

A ⋃ B contains all elements in either set.

A ⋂ B contains only those elements in both sets—in the overlap of the circles.

A^c will incorporate all elements non in the set A. A^c ⋂ B will contain the elements in set B that are non in prepare A.

Case ten

Apply a Venn diagram to illustrate (H ⋂ F) ^c  ⋂ Westward

We'll start by identifying everything in the set H ⋂ F

Now, (H ⋂ F) ^c  ⋂ W will comprise everything not in the set identified above that is too in set West.

Example 11

Create an expression to represent the outlined role of the Venn diagram shown.

The elements in the outlined set are in sets H and F, but are non in set Due west. So we could stand for this prepare every bit H ⋂ F ⋂ W ^c

Try It Now

Create an expression to stand for the outlined portion of the Venn diagram shown

Cardinality

Oft times nosotros are interested in the number of items in a set or subset. This is called the cardinality of the set.

Cardinality

The number of elements in a gear up is the cardinality of that fix.

The cardinality of the fix A is often notated equally |A| or north(A)

Instance 12

Let A = {1, 2, 3, 4, v, 6} and B = {2, 4, half dozen, eight}.

What is the cardinality of B? A ⋃ B, A ⋂ B?

Answers

The cardinality of B is 4, since there are 4 elements in the set.

The cardinality of A ⋃ B is seven, since A ⋃ B = {i, ii, 3, 4, 5, 6, eight}, which contains 7 elements.

The cardinality of A ⋂ B is 3, since A ⋂ B = {2, 4, 6}, which contains iii elements.

Example 13

What is the cardinality of P = the prepare of English names for the months of the year?

Answers

The cardinality of this set is 12, since there are 12 months in the twelvemonth.

Sometimes we may be interested in the cardinality of the matrimony or intersection of sets, just not know the bodily elements of each fix. This is common in surveying.

Example fourteen

A survey asks 200 people "What beverage do you drink in the morning", and offers choices:

• Tea but
• Coffee only
• Both java and tea

Suppose twenty report tea just, eighty study coffee only, twoscore report both.   How many people drink tea in the morn? How many people drink neither tea or coffee?

Answers

This question can well-nigh hands be answered by creating a Venn diagram. We tin run across that we can detect the people who potable tea by adding those who potable merely tea to those who drinkable both: 60 people.

We can also see that those who potable neither are those not contained in the whatever of the three other groupings, and then we tin count those by subtracting from the cardinality of the universal set, 200.

200 – 20 – fourscore – xl = threescore people who potable neither.

Example 15

A survey asks: "Which online services have you used in the last calendar month?"

• Twitter
• Facebook
• Accept used both

The results prove 40% of those surveyed have used Twitter, 70% accept used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?

Answers

Let T exist the set of all people who take used Twitter, and F be the fix of all people who have used Facebook. Notice that while the cardinality of F is 70% and the cardinality of T is twoscore%, the cardinality of F ⋃ T is not simply seventy% + 40%, since that would count those who utilise both services twice. To discover the cardinality of F ⋃ T, we can add the cardinality of F and the cardinality of T, then decrease those in intersection that we've counted twice. In symbols,

northward(F ⋃ T) = n(F) + n(T) – n(F ⋂ T)
north(F ⋃ T) = seventy% + xl% – 20% = xc%

Now, to find how many people take non used either service, we're looking for the cardinality of (F ⋃ T)c . Since the universal set up contains 100% of people and the cardinality of F ⋃ T = 90%, the cardinality of (F ⋃ T)c must be the other ten%.

The previous example illustrated two of import properties

Cardinality properties

n(A ⋃ B) = n(A) + n(B) – due north(A ⋂ B)

north(Ac) = due north(U) – n(A)

Discover that the beginning holding can likewise be written in an equivalent form by solving for the cardinality of the intersection:

n(A ⋂ B) = north(A) + n(B) – north(A ⋃ B)

Case xvi

Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the adjacent quarter.

21 were taking a SS form	26 were taking a HM course
19 were taking a NS form	9 were taking SS and HM
7 were taking SS and NS	ten were taking HM and NS
3 were taking all three	7 were taking none

How many students are just taking a SS class?

Answers

It might help to look at a Venn diagram. From the given data, we know that at that place are three students in region due east and 7 students in region h.

Since 7 students were taking a SS and NS course, nosotros know that northward(d) + n(e) = seven. Since we know in that location are 3 students in region 3, there must be 7 – 3 = 4 students in region d.

Similarly, since there are x students taking HM and NS, which includes regions e and f, in that location must exist 10 – 3 = vii students in region f.

Since 9 students were taking SS and HM, in that location must be ix – 3 = vi students in region b.

Now, nosotros know that 21 students were taking a SS course. This includes students from regions a, b, d, and east. Since we know the number of students in all but region a, we tin can make up one's mind that 21 – six – four – 3 = 8 students are in region a.

viii students are taking but a SS course.

Try Information technology At present

Ane hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.

43 believed in UFOs	44 believed in ghosts
25 believed in Bigfoot	x believed in UFOs and ghosts
8 believed in ghosts and Bigfoot	5 believed in UFOs and Bigfoot
2 believed in all three

How many people surveyed believed in at to the lowest degree one of these things?

How To Find Elements In A Set,

Source: https://courses.lumenlearning.com/atd-hostos-introcollegemath/chapter/set-theory/

Posted by: lealpolornet.blogspot.com

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