SETS

Prepared by CARLOS  I.  GIL,  Department of Mathematics,  Miami Dade College, North Campus

 

We will just mention that a set is a collection of objects that share a common “well-defined” property P.  We say “well-defined” in the sense that if we are given a collection, we can tell with one hundred percent certainty whether a given object belongs to the collection or not. 

 

Example: the collection of lowercase letters of the alphabet from “c” to “g.”  If we examine the object “d,” we can tell with one hundred percent certainty that “d” belongs to the given collection.  If we now examine the object “q,” we can tell with one hundred percent certainty that “q” does not belong to the given collection.  We can do the same for any given object and we are able to tell with one hundred percent certainty, whether the object belongs to the given collection or not.  Therefore, the given collection contains objects that have a common “well-defined” property, so it is a set. 

 

Sets can be specified by a rule of formation, by listing the elements, or by set builder notation.

 

Rule of Formation: a word description of the objects in a particular set. 

Example:   In the above example, the description of the objects was done by the rule of formation: “lowercase letters of the alphabet from c to g.”

 

Listing the Elements, (also known as Roster Form): the objects that belong to the set are listed, separated by commas, within set braces, {  }.

Example:  {c, d, e, f, g} is the roster form of the “lowercase letters of the alphabet from c to g.”

 

Set Builder Notation:  a symbolic form with structure {x | P}, where x is a variable – a placeholder – that represents the objects of the set.  We read the symbolic form {x | P} as: “the set of elements x with property P.”

Example:  {x | x is a lowercase letter of the alphabet from c to g} is the set builder notation of the set {c, d, e, f, g}

                   Read as: “The set of elements x with the property that x is a lowercase letter of the alphabet from c to g.”

 

Elements

The objects that belong to a particular set are called elements or members.

Example:  Write in roster form the elements of the set {x | x is a month of the year beginning with the letter J}.

                  Solution:  {January, June, July}

 

It is customary to write:  5 Î {1, 2, 3, 4, 5, 6} instead of writing the sentence “the element 5 belongs to the set {1, 2, 3, 4, 5, 6}.”  The symbol Î means “belongs to” or “is an element of” or “is a member of” or “is in.”  Also, we use capital letters from the alphabet to “name” sets, so that we do not have to write out the entire roster every time we refer to a particular set.  For example, we will choose the symbol A to represent the set {1, 2, 3, 4, 5, 6}.  So, we write A = {1, 2, 3, 4, 5, 6}. 

 

Example:  If A = {1, 2, 3, 4, 5, 6}, the 2 Î A means that 2 is in the set {1, 2, 3, 4, 5, 6} or 2 belongs to the set {1, 2, 3, 4, 5, 6} or 2 is a member of {1, 2, 3, 4, 5, 6} or 2 is an element of {1, 2, 3, 4, 5, 6}.  Similarly, 8 Ï A means that 8 is not in the set {1, 2, 3, 4, 5, 6} or 8 does not belong to {1, 2, 3, 4, 5, 6} or 8 is not a member of the set {1, 2, 3, 4, 5, 6} or 8 is not an element of the set {1, 2, 3, 4, 5, 6}.

 

Many sets contain an infinite number of elements that can be written in roster form by listing the first few elements followed by three dots ….  The first few elements must be sufficient to identify a pattern that tells us how to obtain the subsequent elements.

 

Example: the set of counting numbers {1, 2, 3, 4, …}.  The three dots indicate that the list continues indefinitely with the same pattern. 

 

Other sets contain a finite number of elements, but the list of elements might be too long to write.  For example, the set of counting numbers from 1 to 100 can be written in roster form as follows:  {1, 2, 3, 4, … , 100}. 

Lastly, many other sets are difficult to write in roster form, so we use set builder notation.  For example, the set of rational numbers:  

Also, the set of real numbers in the interval from 1 to 4 is written:  {x | 1 £ x £ 4}.  And so on…

 

Empty Set

A set with no elements is said to be empty.  An example of an empty set is the set of counting numbers between 1 and 2.  Another example of an empty set is the set of months of the year that begin with the letter X. 

The symbol Æ is used to represent the empty set.  Also, a pair of braces { } with nothing in between is sometimes used to represent the empty set.  The symbols {Æ} or {0} do not represent empty sets.  Why?

 

Subsets

Consider the sets:  A = {60, 70, 90}and B = {50, 60, 70, 80, 90}.  Notice that every element of set A is also an element of set B.  We say that set A is a subset of set B.  Consider the three-month set:  T = {May, June, July}.  Notice that every month in set T is also a month of the year from January to December.  Therefore, we say that the three-month set T is a subset of the set of months of the year.

 

More generally:  Suppose we are given two nonempty sets, A and B.  If every element of set A is also an element of set B, then set A is said to be a subset of set B.

We write the symbols A Í B instead of writing “set A is a subset of set B.”  The symbol Í means “is a subset of” and includes the possibility of A and B being equal.  If we know that the given sets A and B are not equal, then we write A Ì B, and we mean that A is a proper subset of set B.

Remark:  Given a set A, the empty set Æ and set A itself are subsets of A.  They are said to be improper subsets of A.  Check that they satisfy the definition of subsets.

 

Example:  Write all possible subsets of the set  S = {a, b, c}.

                   Solution:  The subsets are:  Æ,  {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.

                                      Notice that there are 8 subsets and set S contains 3 elements.

 

In general, if a set S contains a finite number of elements n, where n is a whole number, then there are  subsets of S.

Example:  Set  S = {a, b, c} contains n = 3 elements.  Therefore, there are 23 = 8  subsets of set S.

 

Equality of Sets

Two nonempty sets, A and B, are said to be equal if they contain exactly the same elements.

Examples:  1)  If  A = {2, 3, 5}  and  B = {5, 2, 3},  then  A = B.  Notice that the order in which the elements are listed makes no difference.

                    2)  Suppose that  S = {90, 3, 1, 75, x, 2}  and  T = {3, 75, 62, 2, 90, 1}.  If  S = T,  find the value of x.  (This is a good exercise).

 

In general, if A and B are two nonempty sets, then A = B if and only if  A Í B  and  B Í A.

 

We use the symbol  Ë  to mean “is not a subset of.” 

Example:   If  A = {5, 61, 75} and B = {5, 47, 75, 83}, then  A Ë B because not all the elements of A are in B.  Similarly,  B Ë A.  Why?

 

Universal Set

Every set can be considered to be a subset of some larger set called Universal Set.  The symbol for the universal set is U.

Examples:  The above mentioned set {c, d, e, f, g} has universal set U = {the set of all lowercase letters of the alphabet}.

For the set  {1, 2, 3, 4, 5, 6} a universal set U = {the set of all whole numbers 1, 2, 3, 4, …}

For the set  {January, June, July} a universal set U = {the set of all months of the year from January to December}.

If we are given a finite number of sets, we could also “construct” our own universal set as long as it contains at least all of the elements being considered.

For example, for the given sets A = {0, 1, 5},  B = {1, 3, 4, 6}, we can use the universal set U = {0, 1, 2, 3, 4, 5, 6}.  Why?

 

OPERATIONS WITH SETS

We will describe briefly the operations of union, intersection, difference, and complements of sets.

Suppose that A and B are sets.

Union:  Denoted A È B, consists of the elements that are in set A or in set B. 

Symbolically,  we write:  A È B = {x | x Î A  or  x Î B}

Intersection:  Denoted A Ç B, consists of the elements that are in set A and in set B. 

Symbolically, we write:  A Ç B = {x | x Î A  and  x Î B}

Difference:  Denoted A – B, consists of the elements that are in set A but not in set B. 

Symbolically, we write:  A - B = {x | x Î A  and  x Ï B}

Complement:  Denoted A’, consists of the elements that are in the universal set U but not in set A. 

Symbolically, we write:  A’ = {x | x Î U  and  x Ï A}

 

Notice that  A’ = U – A;  A È Æ = A;  A Ç Æ = Æ;  A È U = U;  A Ç U = A;  U’ = Æ;  and  Æ’ = U.

 

EXAMPLES

Use the sets  S = {90, 93, 97, a, b};  T = {99, c};  V = {93, 97, 99, a, b, c}; 

with universal set U = {90, 91, 93, 95, 97, 99, a, b, c, d} to perform the indicated operations and answer the questions.

 

1)          S È V = {90, 93, 97, 99, a, b, c}

2)          S Ç V = {93, 97, a, b}

3)          S È T = {90, 93, 97, 99, a, b, c}

4)          S Ç T = Æ

5)          S¢ = {91, 95, 99, c, d}

6)          S¢ Ç T = {99, c}

7)          S – V = {90}

8)          V – S = {99, c}

9)          Is it true that T Ì S ?   YES.   (Why?)

10)      Is it true that S Ì T ?    NO.   (Why?)

 

 

EXERCISES

In exercises 1–4, write the given set in roster form.  If the set is empty, so state.

1)   {x | x is an even counting number between 3 and 9}

2)   {y | y is a day of the week beginning with the letter T}

3)   {x | x is a day of the week beginning with the letter B}

4)   {y | y + 2 = 5}

 

In exercises 5–6, write the given set in set builder notation.

5)   {1, 2, 3, 4, 5, 6}

6)   {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

 

In exercises 7–35, use the sets  A = {1, x, 5, 2};  B = {x, y, 1, 4, 6};  C = {1, 6};  with universal set  U = {0, 1, 2, 3, 4, 5, 6, w, x, y, z}, to perform the indicated operations and answer the questions.  If the answer is the empty set, so state.

7)    B’                                                                         8)    A È B

9)    A Ç B                                                                   10)  A’

11)  A – B                                                                    12)  A Ç B’

13)  Based on your answers to problems 11 and 12, what can you say about A – B and A Ç B’ ?

14)  B – A                                                                    15)  A’ Ç B

16)  Based on your answers to problems 14 and 15, what can you say about B – A and A’ Ç B ?

17)  A’ È B’                                                                 18)  (A Ç B)’

19)  Based on your answers to problems 17 and 18, what can you say about A’ Ç B’ and (A È B)’ ?

20)  (A È B)’                                                               21)  A’ Ç B’

22)  Based on your answers to problems 20 and 21, what can you say about A’ È B’ and (A Ç B)’ ?

23)  B È C                                                                   24)  B Ç C

25)  B – C                                                                    26)  C - B

27)  Based on your answers to problems 11 and 14, what can you say about A – B and B – A ?

28)  Is it true that A Ì B ?                                             29)  Is it true that B Í A ?

30)  Is it true that B Í C ?                                            31)  Is it true that C Ì B ?

32)  Is true that C Ì C ?                                               33)  Is it true that B Í B ?

34)  Write out all possible subsets of set A.                   35)  How many subsets does set B have?

 

 

SETS OF NUMBERS

We assume the existence of the following sets of numbers:

Natural Numbers:  (Also called Counting Numbers or Positive Integers),   N = {1, 2, 3, 4, …}

Whole Numbers:  (Also called Non-negative Integers),   W = {0, 1, 2, 3, 4, …}

Integers:   Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Rational Numbers:   Q =  

Examples:  The following are examples of rational numbers

,  ,  0.7,  0.358,  1.333333…,  ,  0.81818181…,  0,  -4,  1,  ,  , 

Rational numbers include all natural numbers, all whole numbers, all integers, all decimals with a terminating decimal part or with a non-terminating and repeating decimal part, all perfect powers (perfect squares, perfect cubes, …), and all fractions of the form , with p and q being integers and q ¹ 0.

 

Irrational Numbers:   Irr = {x | x Ï Q}

Examples:  The following are examples of irrational numbers

0.717117111711117111117…,  0.123456789101112…,  2.753192035472…(no discernible pattern),  ,  ,  ,  ,  ,  .

Irrational numbers include all numbers that are not rational numbers.  That is, irrational numbers cannot be any whole number, any integer.  The decimal numbers must have a non-terminating and non-repeating decimal part.  Irrationals include also roots of numbers that are not perfect powers and they are numbers that cannot be expressed as a fraction of the form , with p and q being integers and q ¹ 0.

 

Real Numbers:   Â = Q È Irr

The set of real numbers includes all counting numbers, all whole numbers, all integers, all rational numbers and all irrational numbers.

 

EXERCISES

1)  Consider the following set of numbers:

 

a)      List the natural numbers that appear in the given set

b)      List the whole numbers

c)      List the integers

d)      List the rational numbers

e)      List the irrational numbers

f)       List the real numbers

 

2)  Give an example of an irrational number not yet listed above.

 

3)  Is the number 27.123456789101112  rational or irrational?

 

4)  Is the number  rational or irrational?