# Solutions: Sets and Set Theory

#### If you would like to better understand the curriculum, here are the learning objectives for this unit.

#### Introduction to Sets

There are four suits in a standard deck of playing cards: hearts, diamonds, clubs and spades.

C is the set of whole numbers less than 10 and greater than or equal to 0. Set *D* is the even whole numbers less than 10, and set *E* is the odd whole numbers less than 10.

Set *G* is the set of all oceans on earth. Set *E* is a set of some rivers, and set *F* is a list of continents.

Set *Z* is the set of all types of matter. Set *X* is a set of some metals and set *Y* is a set of some gases.

Set *A* lists the element r twice. So the objects in this set are not unique.

#### Basic Notation

Liquid is an element of set *R*.

The number 7 is an element of set *G*.

The colors red, white and blue are all colors of the US flag, and are all elements of set *B*.

A bobcat was not listed in set *X*.

All of these territories are outside of the United States.

#### Types of Sets

Set *G* has a finite number of elements.

Set *H* is the set of integers, which has an infinite number of elements.

Each set listed in Exercise 3 is a finite set.

The integers is the only set in Exercise 4 that is infinite.

There are no cars with 20 doors, so this set is empty (null).

#### Set Equality

Since *P* and *Y* contain exactly the same number of elements, and the elements in both are the same, we say that *P* = *Y.*

Set *Q* contains the element 7, which is not an element of set *H*. Thus *H* ≠*Q*

*M* = {0, 2, 4, 6, 8, 10} and N = {0, 2, 4, 6, 8}. Therefore *M* ≠*N.*

Since *X* and *Y* contain exactly the same number of elements, and the elements in both are the same, we say that *X* = *Y.*

Since *A* and *B* are both empty sets, we say that *A* = *B.*

#### Venn Diagrams

*A* = {2, 4, 6, 8} and the range given for A is non-inclusive.

Choice 1 uses the wrong notation, choice 2 is correct, choice 3 is wrong, and choice 4 is wrong.

This Venn diagram represents the intersection of *P* and *Q, *which is 6.

The elements* *2, 3, 5, 7, and 11 are all elements of *X*.

The union of *X *and* Y *is shown in this Venn diagram, by the shaded region.

#### Subsets

Sets *X, Y, *and* Z *are each subsets of set* G.*

Every element of the set of vowels is contained in the set of the alphabet*.*

Since 9 is not an element of *A, *we know that* C *is not a subset of* A*.

There are 5 elements in set T, so the number of subsets of T is 25, which equals 32.

Set *R *= {0, 1, 2, 3, 4} and set* S *= {4, 3, 0, 2, 1}, thus *R* is equivalent to *S.*

#### Universal Set

Each of the elements in set *G *and* *set *H *are integers. None of these elements are fractions nor irrationals.

By definition, sets *X *and *Y *are each subsets of the universal set. So the world must be the universal set.

By definition, sets *M *and *N *are each subsets of the universal set. The intersection of *M *and *N *is null. So all of the above is the correct answer.

Triangles does not overlap with quadrilaterals, but triangles is a subset of the universal set (polygons).

Factors of 36 overlaps with set *P*, and is a subset of the set of whole numbers less than 40 (the universal set).

#### Set-Builder Notation

The set of all q such that q is an integer greater than or equal to ^{–}4 and less than 3.

The set of all x such that x is a real number greater than or equal to 4.

Each set listed is equal to “the set of all n such that n is an integer less than 2”.

Set builder notation is usually used with infinite sets. Choices 1 and 2 are each finite sets that do not have numbers as their elements. By process of elimination, choice 3 makes sense.

The elements given are real numbers. None of the choices (1-3) are sets with real-number elements.

#### Complement of a Set

The complement of a set is the set of elements which belong to but which do not belong to *A*.

*X** ‘ *= {

*n | n*

*and n*

*X*}

The complement of set *P* is the set of elements which belong to but which do not belong to *P*.

The complement of set *N* is the set of elements which belong to (the alphabet) but which do not belong to *N*.

Draw a Venn diagram to help you find the answer.

The complement of set *A* is the set of elements which belong to but which do not belong to *A*.

*A** ‘ *= {

*x | x*

*and x*

*A*}

#### Intersection

Oranges and pears are common to both sets.

The number 2 is the only even prime.

These sets have no elements in common.

The number 3 is an element of *P*, and is not in the intersection of *P* and *Q*.

#### Union

*A* * B = {x | x*

*A*or x

*B*}The numbers 0 and 1 are neither prime nor composite.

The union of a set and its complement is the Universal Set.

#### Practice Exercises

Choice 1 uses set-builder notation, choice 2 describes the set, and choice 3 uses roster notation.

The element ^{–}2 is not an element of set *D*.

All of the sets listed are finite except choice 4.

*X* = *Y* since *X* and *Y* contain exactly the same number of elements, and the elements in both are the same.

The element n is not a member of set *P*. So set *S* cannot be a subset of set *P*.

Both triangles ate trapezoids are subsets of polygons.

Choice 2 is the set of q such that q is an element of the integers, and q is greater than or equal to ^{–}5.

*A** ‘ *= {

*x | x*

*and x*

*A*}

The shaded region shows a union of sets *X* and *Y*.

The shaded region shows an intersection of sets *X* and *Y*.

#### Challenge Exercises

The letters m, a and t are listed more than once, so the objects are not unique.

9 is not an element of *C*.

Choice 2 is a finite set; the rest are infinite.

Choice 3 is equal to the set of n such that n is an element of the integers, and n is greater than or equal to ^{–}3 and less than 7.

There are 5 elements in M, so the number of subsets is 25 which equal 32.

*P* = {2, 3, 5, 7, 11, 13, 17, 19} and *Q* = {2, 4, 6, 8, 10, 12, 14, 16, 18}. The element 2 is in the intersection of both sets.

Choice 2 correctly describes the set given in set-builder notation.

Complement of a set is defined as:* A** ‘ *= {

*x | x*

*and x*

*A*}