Download The language and symbols of math.

About the course:
The language and symbols of math. Math is an exact science in
the sense that it is not subjective, but objective. A statement is either
true or false. Solutions and proofs are either correct or incorrect. There
is room for variation in the amount of detail that is given in the proof.
We must communicate mathematics in verbal and written form both
formally and informally. This is not easy to learn and there are many
aspects to that process, which can be confusing. If you see the symbol
R in math context, you are to understand it to mean the real numbers.
There are conventions, like using the letter x to indicate a real number.
However, it is never assumed that x is a real number. To indicate that,
you would say, “Let x be a real number”, or “Let x ∈ R”. Another
convention is to use the letter n to indicate an integer. It would not be
wrong to say, “Let n ∈ R”, but it may make it a little harder on the
reader who is used to having
n represent integers.
P
We use the symbol
to indicate a sum, which can be indexed in
many different ways. Examples include:
n
X
i=1
,
X
j∈I
,
X
.
i6=k
You may use the same symbol to mean something else, but inPthat
case, the meaning must be clearly stated. One might say, “Let
be
the set of allPtrigonometric functions”. We would say that the default
meaning of
is that of a sum.
We have words in math that mean different things in different contexts. The word “tangent” can be used to mean a trigonometric function, or a type of line. We have symbols that mean different things in
different contexts. Using “−1” as a superscript could mean the power
of negative 1 or the inverse.
We have learned about addition from our early years as a child. After
examination, we realize that addition of real numbers has properties of
closure, associativity, commutativity, an identity, and each number
has a negative. Now we may use the symbol + to indicate an operation that has these same properties, but may be different from our
usual understanding of the operation of addition that we learned in
elementary school.
To make matters worse, we love to use shorthand notation. If I want
to indicate several sets,
{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {3}, {2}, ∅,
1
2
I might get tired of writing out all of the set brackets, { }, and just
write:
123, 12, 13, 23, 1, 3, 2, ∅.
If I verbally explain this to the observer, the observer should know
what I am talking about, but when one just looks at the second line,
it could be shorthand for any of several different things. Some sentences have commonly used shorthand. For instance, “We will show
the implication” is often shortened to “(⇒)” and “The following are
equivalent” to “Tfae”.
These types of abbreviations are not found in many formal writings
and should be avoided in your final portfolio. This course will get
you started on the right path and sets the stage with the basics. I
expect you to learn and practice the formal style of writing and verbal
communication, as you would expect to find in textbooks and research
articles.
Note about typesetting. The notation for everything you hand in
needs to be proper. For instance, when entering a formula on a graphing calculator, you might write: (2∧ p−2)\2∧ p. But this is a translation
of the correct notation, which is:
2p − 2
2p
Subscripts need to be subscripted. For instance, if you mean A1 , A2 ,
. . . , An , do not write A1, A2, . . . , An.
Our goal is to learn proper symbolism and practice using it. So
that when you see a symbol or term used in context, you learn to
immediately recognize and understand it.
You must first know what you are abbreviating before the abbreviations can make any sense.
Proving theorems. A mathematical statement is either true or false.
In this course, you will learn many standard proof techniques. Logic is
the foundation of all of these techniques. Through precise definitions
of words like and, or, implies, if , not, every, exists, and equivalent
we will develop valid methods of proofs. Then we will practice all
of these proof techniques to prove many theorems. In doing so, we
will cover much of the foundational material that is used in every subject in mathematics. When you complete this course, you will know
proper definitions of terms that you have used before, like f unction
and limit and definitions of common terms that might be new to you
like permutations and contrapositive.
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Topic: Theorems in mathematics are based on facts, axioms,
and definitions. We will start with something familiar.
The Pthagorean Theorem.
Theorem 1. Let T be a right triangle with hypotenuse of length c and
two other sides of lengths a and b. Then c2 = a2 + b2 .
Proof. Consider a square of side length a + b. On each side, mark a
point between a segment of length a and one of length b in such a way
that connecting the point on one side to the point on the next adjacent
side gives a right triangle with legs of length a and b and hypotenuse
c. After joining the 4 such pairs of points we see that the square has
a square of side length c inscribed in it. (How do we know this is a
square?) We see that the area of the square can be computed as either
(a + b)2 or c2 + 2ab. Thus,
(a + b)2 = c2 + 2ab
and so
a2 + b2 = c2 .
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Name some integer solutions.
Fermat’s Last Theorem.
Theorem 2. Let n ≥ 3 be a natural number. There exist no triples of
integers, a, b, and c such that
an + bn = cn .
Conjecture. A conjecture is a mathematical statement, the truth
value of which has not yet been determined. It is thought to be true
by the person who first published it.
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Sections 1.1 and 1.2. A set is a collection of objects called elements. We usually use capital letters to denote sets.
x∈A
x 6∈ A
A=B
A⊆B
-
x belongs to A
x is not an element of A
every element of A is in B and every element of B is in A.
every element of A is in B, A is a subset of B, B contains A.
So A = B if and only if A ⊆ B and B ⊆ A.
A ⊂ B - A ⊆ B, but A 6= B. B is a proper subset of A.
A 6⊆ B - A not contained in B. There is some element of B that is not
in A.
∅ - The empty set. The set that has no elements. It is a subset
of every set, even of itself.
The power set of a set A is denoted, P(A), and is the set which
contains all subsets of A.
Naturals,
Wholes,
Integers,
Cardinality of a finite set,
collection of sets = family of sets.
Indexed family of sets.
Roster method of indicating a set.
Set builder notation way of indicating a set.
* S = {x ∈ R : x > 7}
* T = {x ∈ R : a ≤ x ≤ b}
• Interval notation, for a, b ∈ R, a < b:
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(a, b) [a, b] [a, b) (a, b] (a, +∞) (−∞, b) [a, +∞) (−∞, b]
•
•
•
•
•
[n], n ∈ N
Evens
Odds
Well-defined
Venn-diagram
There are many correct ways to denote a given set
The set of odds equals {2k + 1 : k ∈ Z} and it also equals {2k − 1 :
k ∈ Z}. We give a proof.
Proof. Let A be the set of integers expressible as 2k − 1 for some k ∈ Z.
So A = {2k − 1 : k ∈ Z}. Let B be the set of integers expressible as
2s + 1 for some s ∈ Z. So B = {2s + 1 : s ∈ Z}. Let x ∈ A. Then we
know that x = 2k−1 for some k ∈ Z. Then x = 2k−2+1 = 2(k−1)+1.
Since s = k − 1 ∈ Z, we have that x ∈ B. Therefore, A ⊆ B.
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Let x ∈ B. Then we know that x = 2s + 1 for some s ∈ Z. Then
x = 2s + 2 − 1 = 2(s + 1) − 1. Since k = s + 1 ∈ Z, we have that x ∈ A.
Thus, B ⊆ A.
Therefore, A = B.
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Unions, intersections, and compliments.
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the union of A and B, A ∪ B
the intersection of A and B, A ∩ B
the complement of A, denoted Ac
a universal set
the cartesian product of A and B, denoted A × B. Also called
the cross product.
S1 × S2 × · · · × Sn .
2 sets are called disjoint if their intersection is empty.
A family of sets is called mutually disjoint or pairwise disjoint if every pair of sets in the family is disjoint.
Two sets are called nested if one is contained in the other.
A family of sets is called nested if given any two, one is contained in the other.
set difference, denoted A\B, or A − B,
Distributive Laws: (A∪B)∩C = (A∩C)∪(B ∩C) and (A∩B)∪C =
(A ∪ C) ∩ (B ∪ C).
Proof:
Theorem 3. (1.2.1:) S, T finite then
a) |S ∪ T | = |S| + |T | − |S ∩ T |,
b) if S and T are disjoint then |S ∪ T | = |S| + |T |,
c) if S ⊆ T , then |S| ≤ |T |.
Proof: Later.
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Example 1. Let U = {0, 1, . . . , 10}, A = {0, 1, 2, 9}, and B = {2, 3, 4, 5, 6}.
Describe in roster notation
A ∩ B, A ∪ B, Ac , A − B, A\B, B c , Ac ∪ B c , (A ∩ B)c
Example 2. Use interval notation to describe:
[−3, 7] ∩ (5, 9], [−3, 7] ∪ (5, 9], [−3, 7] − (5, 9]
{x ∈ R : |x| ≤ .01}, {x ∈ R| |x| > 2}
De Morgan’s Laws. The complement of union is the intersection of
individual complements and
The complement of intersection is the union of individual complements.
Theorem 4. (De Morgan’s Laws:) A, B sets
a) (A ∪ B)c = Ac ∩ B c ,
b) (A ∩ B)c = Ac ∪ B c .
Proof:
The general version of De Morgan’s Laws will be given later.
Theorem 5. (1.2.6:)If A and B are nonempty finite then |A × B| =
|A| · |B|.
The general version of Theorem 1.2.6 will be given later. (Theorem
1.2.9)
Tree diagrams