# Solutions: Sets and Set Theory

#### 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 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 = {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 and set are integers. None of these elements are fractions nor irrationals.

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

By definition, sets and are each subsets of the universal set. The intersection of and 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.

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 | xA or xB}

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 }