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 }
