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MATH 111: FOUNDATIONS OF MATHEMATICS
1
...
This procedure is the most
basic motivation for learning the whole numbers and learning how to add and subtract
them
...

Such counting can to situations that may at first seem contradictory
...
This seems to be impossible since June only has 30 days
...
Let W be the set
of windy days, and R be the set of rainy days
...
Then W
and R together have size 25, so the overlap between W and R is 10
...


We can use these sets understand relationships between groups, and to analyze survey
data
...

Describing a set
A set can be described by listing all of its elements
...

The five elements of the set are separated by commas, and the list is enclosed between
curly brackets
...
Thus the set S above can also be written as

S = { odd whole numbers less than 10 }, which we read as ‗S is the set of odd whole
numbers less than 10‘
...
This means that our description of the elements of a set is
clear and unambiguous
...
An example of a well-defined set is T = { letters in the
English alphabet }
Equal sets
Two sets are called equal if they have exactly the same elements
...
This is written as { 1, 3, 5 } ≠ { 1, 2, 3 }
...

For example, { 1, 3, 5, 7, 9 } = { 3, 9, 7, 5, 1 } = { 5, 9, 1, 3, 7 }
...
For example, { a, a, b } = {
a, b }
...
The second mention of a is an
unnecessary repetition and can be ignored
...

The symbols  and 
The phrases ‗is an element of‘ and ‗is not an element of‘ occur so often in discussing sets
that the special symbols  and  are used for them
...
)
8  A (Read this as ‗8 is not an element of the set A‘
...


• A set must be well defined, meaning that its elements can be described and listed
without ambiguity
...

• Two sets are called equal if they have exactly the same elements
...
- Any repetition of an element is ignored
...

• If b is not an element of a set S, we write b  S

EXERCISE 1
a) Specify the set A by listing its elements, where A = { whole numbers less than 100
divisible by 16 }
...

c) Does the following sentence specify a set? C = { whole numbers close to 50 }
...

Here are two more examples:
{ whole numbers between 2000 and 2005 } = { 2001, 2002, 2003, 2004 }
{ whole numbers between 2000 and 3000 } = { 2001, 2002, 2003,…, 2999 }
The three dots ‗…‘ in the second example stand for the other 995 numbers in the set
...
This
notation can only be used if it is completely clear what it means, as in this situation
...
Here are two
examples of infinite sets:
{ even whole numbers } = { 0, 2, 4, 6, 8, 10, …}
{ whole numbers greater than 2000 } = { 2001, 2002, 2003, 2004, …}

Both these sets are infinite because no matter how many elements we list, there are
always more elements in the set that are not on our list
...

The numbers of elements of a set
If S is a finite set, the symbol | S | stands for the number of elements of S
...

If A = { 1001, 1002, 1003, …, 3000 }, then | A | = 2000
...

The set S = { 5 } is a one-element set because | S | = 1
...

The empty set
The symbol  represents the empty set, which is the set that has no elements at all
...

|  | = 0 and x   , no matter what x may be
...


Finite and Infinite sets
• A set is called finite if we can list all of its elements
...

• If S is a finite set, the symbol | S | stands for the number of elements of S
...
Thus |  | = 0
...


EXERCISE 2
a) Use dots to help list each set, and state whether it is finite or infinite
...

i) S = { primes }
ii) S = { even primes }
iii) S = { even primes greater than 5 }
iv) S = { whole numbers less than 100 } c Let F be the set of fractions in simplest form
between 0 and 1 that can be written with a single-digit denominator
...

SUBSETS AND VENN DIAGRAMS
Subsets of a set
Sets of things are often further subdivided
...
We express this in the language of sets by saying that
the set of owls is a subset of the set of birds
...
This is
written as
S  T (Read this as ‗S is a subset of T‘
...

Thus { owls }  { birds } because every owl is a bird
...

The sentence ‗S is not a subset of T‘ is written as S ⊈ T
...
For example,
{ birds } ⊈ { flying creatures } because an ostrich is a bird, but it does not fly
...

The set itself and the empty set are always subsets
Any set S is a subset of itself, because every element of S is an element of S
...

Furthermore, the empty set  is a subset of every set S, because every element of the
empty set is an element of S, there being no elements in  at all
...

Every element of the empty set is a bird, and every element of the empty set is one of
the numbers 1, 2, 3, 4, 5 or 6
...

For example,
{ owls }  { birds } means ‗All owls are birds
...
‘
{ rectangles }  { rhombuses } means ‗ all rectangles are rhombuses
...
For example,
{ owls }  { birds } means ‗If a creature is an owl, then it is a bird
...


A proper subset is a subset that is not identical to the original set—it contains fewer
elements
...
Alternatively we say B is a proper subset of C since all elements of B
belong to C and C has at least one more element than B
...

Algebra of sets (Union, Intersection, and Complement)
Suppose you and a new roommate decide to have a house party, and you both invite
your circle of friends
...
This is a good example of how
sets interact
...

The union of two sets contains all the elements contained in either set (or both sets)
...
More formally, x ∊ A ⋃ B if x ∈ A or x ∈ B (or both)
The intersection of two sets contains only the elements that are in both sets
...
More formally, x ∈ A ⋂ B if x ∈ A and x ∈ B
...
The complement
is denoted A’, or Ac, or sometimes ~A
...
Find A ⋃ B
2
...
Find Ac⋂ C

Answers
1
...

2
...
Here we‘re looking for all the elements that are not in set A and are also
in C
...
For this reason, complements are usually only used with intersections, or when
we have a universal set in place
...
This would
have to be defined by the context
...

EXAMPLE
1
...

2
...

3
...
If A = {1, 2, 4},
then Ac = {3, 5, 6, 7, 8, 9}
...

Grouping symbols can be used like they are with arithmetic – to force an order of
operations
...
Find (H ⋂ F) ⋃ W
2
...
Find (H ⋂ F)c ⋂ W
Solutions
1
...
Now we union that result
with W: (H ⋂ F) ⋃ W = {dog, duck, rabbit, deer, frog, mouse}
2
...
Now we intersect that result with H: H ⋂ (F ⋃ W) = {dog, rabbit, mouse}
3
...
Now we want to find the
elements of W that are not in H ⋂ F
...
These illustrations now called Venn Diagrams
...
Overlapping areas indicate elements common to
both sets
...


A ⋂ B contains only those elements in both sets—in the overlap of the circles
...
Ac ⋂ B will contain the elements in
set B that are not in set A
...


EXAMPLE
Create an expression to represent the outlined part of the Venn diagram shown
...
So we could
represent this set as H ⋂ F ⋂ Wc

EXERCISE
Create an expression to represent the outlined portion of the Venn diagram shown

CARDINALITY
The number of elements in a set is the cardinality of that set
...

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

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

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


Sometimes we may be interested in the cardinality of the union or intersection of sets,
but not know the actual elements of each set
...

EXAMPLE
A survey asks 200 people ―What beverage do you drink in the morning‖, and offers
choices:


Tea only



Coffee only



Both coffee and tea

Suppose 20 report tea only, 80 report coffee only, 40 report both
...
We can see that
we can find the people who drink tea by adding those who drink only tea to those who
drink both: 60 people
...

200 – 20 – 80 – 40 = 60 people who drink neither
...
How many people have used neither Twitter or Facebook?
Answers
Let T be the set of all people who have used Twitter, and F be the set of all people who
have used Facebook
...
To find the cardinality of F ⋃ T, we can add the
cardinality of F and the cardinality of T, then subtract those in intersection that we‘ve
counted twice
...
Since the universal set contains 100% of people and the
cardinality of F ⋃ T = 90%, the cardinality of (F ⋃ T)c must be the other 10%
...
Alternatively
n(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)
...

21 were taking a SS course

26 were taking a HM course

19 were taking a NS course

9 were taking SS and HM

7 were taking SS and NS

10 were taking HM and NS

3 were taking all three

7 were taking none

How many students are only taking a SS course?
Answers
It might help to look at a Venn diagram
...

Since 7 students were taking a SS and NS course, we know that n(d) + n(e) = 7
...

Similarly, since there are 10 students taking HM and NS, which includes regions e and f,
there must be 10 – 3 = 7 students in region f
...

Now, we know that 21 students were taking a SS course
...
Since we know the number of students in all but region a, we can
determine that 21 – 6 – 4 – 3 = 8 students are in region a
...


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


2
...
}

Natural numbers are closed under addition and multiplication i
...
For example, 2+5=7 and
3x8=24
...
For example 2-6=-4 which is not a counting number
...



Integers

The set of integers is denoted by Z and fully described by the set
Z={…,-3,-2,-1,0,1,2,3,…}
Note that N  Z , since all natural numbers are integers and there are integers that are
not natural numbers
...
The number line is used to conveniently represent
integers
...
The set of integers is closed under
addition, subtraction and multiplication but not division
...
A
n
multiple of an integer k is any integer expressible in the form m
...

For example 21 is a multiple of 3 because 21=7x3
...
e each even integer can be
expressed in the form 2k, where k is an integer
...
0 is an even integer because 0=2x0
...
For example 9=2(4)+1 and 7=2(4)1
EXAMPLE
Prove that if k is any integer then
a) The sum of 2k+1 and 2k-1 is an even integer
b) The product of 2k+1 and 2k-1 is an odd integer
SOLUTION
a) (2k+1)+(2k-1) = 4k = 2(2k)
...

b) (2k+1)(2k-1) = 4k2-2k+2k-1 = 4k2-1 = 2(2k2)-1
...



Prime Numbers

Any positive integer that is not 1 and has exactly two positive factors, 1 and itself
...
Numbers that are neither 1 nor prime
are called composite numbers , e
...
g 5 
that are not integers e
...
Every rational number can be expressed as a decimal, for
7

example
32
 1
...
13333
...
181818
...
321

ii)0
...
212121

SOLUTION
i) 7
...
321 is rational
...
999 then 10m=9
...
999-0
...
e 9m = 9,
hence m=1, therefore 0
...

Attempt part iii) as an exercise
...
e not rational
...


SOLUTION
This proof is by contradiction
...
e that 2 is rational, and let
integers and the fraction

2

a
b

(i), where a and b are

a
is assumed to be fully reduced (i
...
Squaring (i) we get

 2

2

2

a2
a
   i
...
Since a2 is twice the integer b2 then a is even and can be written as a = 2k
...
Since both a and b are even
they have a common factor of 2
...
Therefore it follows that the assumption that
hence

2 is rational is false,

2 is an irrational number
...



Real and Complex Numbers

Real numbers denoted by R are numbers that can be represented by a decimal
...

C = { a  ib | a  R , b  R and i 2  1 }
Each real number is a complex number since a  a  0i , but there are complex numbers
that are not real numbers, hence R  C

POSTULATES OF ALGEBRA
A mathematical system is constructed by adopting assumptions about elements of a set,
operations on elements and the relation between the elements
...
A proof of a statement W is a well organized logical
argument which demonstrates that W follows from the adopted postulates and
theorems proved previously
...
Then the equality relation satisfies the
following four postulates
i
...

iii
...


E1 Reflexive Postulate: a  a
E2 Symmetric Postulate: If a  b then b  a
E3 Transitive Postulate: If a  b and b  c , then a  c
E4 Substitution Postulate: If a  b , then the truth or falsity of a statement is not
changed when any or all occurrences of a are replaced by b
...

SOLUTION
(a) E1 (b) E3 (c) E2 (d) E4

The Field Postulates
The field postulates for the real numbers are:
F1 Closure Postulate for Addition: For any real numbers a and b, a  b is a real number
...

F5 Commutative Postulate for Addition: For any real numbers a and b, a  b  b  a
...

F7 Associative Postulate for Multiplication: For any real numbers a and b, (ab)c = a(bc)
F8 Multiplicative Identity Postulate: There is exactly one real number, 1, such that 1  0 ,
and for every real number a, a 1  a and 1  a  a
F9 Multiplicative-Inverse Postulate: For each real number a other than 0, there is exactly
one real number a 1 such that, a  a 1  1 and a 1  a  1
...

F10 Commutative Postulate for Multiplication: For any real number a and b, ab  ba
...

a(b  c)  ab  ac and (b  c)a  ba  ca
...


EXAMPLE
State one postulate that justifies each statement
a) 5t  0  5t b) 2(nt)  (nt)  2

c) 17  (17)  0

SOLUTION
a) Additive Identity Postulate F3
b) Commutative Postulate for Multiplication F10

d) 4(k  2)  4k  8

c) Additive Inverse Postulate F4
d) Distributive Postulate for Multiplication F11
...
By hypothesis x = a + b, so by postulate E4 3x = 12
Theorem
The following statements in a, b, c, d and e (all not zero) are true and can be derived
from the stated postulates:For any non-zero real number b,

1
 b 1
b

If a = b, then a + c = b + c
If a = b, then a – c = b – c
If a = b, then ac = bc
If a = b, and c  0 , then

a b

c c

-(-a) = a
...
For real numbers a
and b notation aa
a  b is read as ―a is less than or equal to b‖ and is
read as ―b is greater than a‖
...

The order postulates are:

O1 Trichotomy Postulate: For any real numbers a and b, exactly one of the following is
true, ab
...

EXAMPLE
State one order postulate that justifies each of the following statements:
a) If -1<2 and 2<7, then -1<7
b) If -3<1, then 0<4
c) If 6<10 then 3<5
...

2
2

Theorem
For any real numbers a, b and c the following statements are true
a-b
a>b iff –a<-b
If abc
If aIf aIf a>0, c>0, aEXAMPLE
Prove that a-b

SOLUTION
By hypothesis ai
...

EXAMPLE
Solve the following inequalities
(a) 5x ≤ x - 3 (b) 7 < 2x + 3 ≤ 13
SOLUTION
(a) 5x – x ≤ (x – 3) – x, thus 4x ≤ 3 or x ≤ ¾
...

(b) 7 - 3 < (2x + 3)- 3 ≤ 13 – 3, so 4 < 2x ≤ 10, dividing through by two we get: 2 < x ≤ 5
...

The distance between two points on the number line is always indicated by a positive
real number
...
g |-5| = 5 and |0| = 0
...

Let a and b be any two real numbers such that a ...


Interval

Abbreviation

{x| a < x < b}

(a,b)

{x| a ≤ x ≤ b}

[a,b]

{x| a ≤ x < b}

[a,b)

{x| a < x ≤ b}

(a,b]

If S is a non-empty set of real numbers then an upper bound for S is any real number
that is greater than or equal to every member in S
...
For example 7 is the least upper bound for the interval (1,7] but
not (1,7)
...

A lower bound for a non-empty set S of real numbers is any real number that is less
than or equal to every member on S
...


EXAMPLE
Prove that there is no upper bound for the set of Natural numbers N
SOLUTION
The proof is by contradiction
...
Consider any number n in N so that n + 1 is also
in N
...
e, every number n in N is less than or equal to b – 1
...
But that contradicts the fact that b is the least
upper bound for N
...
Hence there is no upper bound for N
EXERCISE
(1) Solve the following inequalities

(a) 0
...
2- 0
...


(c) (3 + x)/6 < 4
...

(a) 2x < 5
...


(d) 5 ≤ x + 3 ≤ 18

(d) 0 ≥ 5x + 1 ≥ -4

(3) Find two rational numbers between 0
...
444…
(4) Prove that one half the sum of any two real numbers is between the two real
numbers
...

(9) Prove that there is no lower bound for the set of integers

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