Download Sets, Sequences, Logic - Lecture Notes | MATH 117 and more Study notes Elementary Mathematics in PDF only on Docsity!
Lecture 2
I. Sets
II. Sequences
III. Logic
SETS
Set : a collection of objects
Element or member: individual object in a set, x ∈ A
Set-builder notation: ex: D = {x | x is an decimal between 0 and 1}
Write in set-builder notation:
Cardinal number of a set, n(X): indicates the number of elements in the set X.
ex: If D = {a, b} then n(D) = 2
Empty set or Null set: set that contains no elements; use symbol Ø or { }
note: {0} does not represent the empty set
Universal set: contains all elements being considered. Notation: U
Venn Diagram: named after John Venn (1834-1923) who used diagrams to illustrate ideas in logic
Complement of F: set of elements in U but not in F
Subset: B is a subset of A, written B ⊆ A, if and only if
every element of B is an element of A.
Proper Subset: means that there is at least one element
of A that is not in B, written B ⊂ A.
Intersection of two sets A and B, written A ∩ B, is the
set of all elements common to both A and B.
A ∩ B = { x | x ∈ A and x ∈ B}
Union of two sets A and B, written A ∪ B, is the set of
all elements in A or B, i.e., A and B combined.
A ∪ B = { x | x ∈ A or x ∈ B}
Cartesian product: written A x B, read A cross B, is the set of all ordered pairs such that the first
component of each pair is an element of A and the second component of each pair is an element of B.
A x B = {(x,y) |x ∈ A and y ∈ B}
F
SEQUENCES
A Sequence is an ordered arrangement of numbers, figures, or objects.
ex 1: 0, 5, 10, 15, 20, 25,...
ex 2: 1, 1, 2, 3, 5, 8, 13, 21,...
ex 3: 2, 6, 10, 14, 18, 22,...
ex 4: 1, 11, 111, 1111, 11111,...
An arithmetic sequence is one in which each successive term is obtained from the previous term by
the addition or subtraction of a fixed number, the difference. Example 1 and 3 above are arithmetic
sequences.
Finding the nth term of an arithmetic sequence:
What is the 200
th term of: 4, 7, 10, 13,...?
What is the nth term of this sequence?
Find the first four terms of a sequence whose nth term is given by 2n + 3.
A geometric sequence is one in which each successive term is obtained from the previous term by
multiplying the predecessor by a fixed number, the ratio.
Ex 1: 2, 4, 8, 16, 32,...
Ex 1: 1, 3, 9, 27, 81,...
Ex 1: 1, 1.5, 2.25, 3.375, 5.0625,...
Can you find the 10
th term of ex 1?
What is the nth term of a geometric sequence?
A Fruity Problem: A group of 10 fruit flies double in a give time period. After 10 such time periods,
how many fruit flies will there be?
Music example: Do the notes of a musical scale form an arithmetic or geometric sequence?
Figurate numbers provide examples of sequences that are neither arithmetic nor geometric.
ex 1: (triangular nos.) 1, 3, 6, 10,...
ex 2: (square nos.) 1, 4, 9, 16,...
Suppose you just won the Urbana Sweepstakes. You have a choice of accepting a lump sum of $
now, or taking $100 per month for life. In either case, you will put the money in a tax-sheltered
savings paying 4% interest, compounded monthly, and let the interest accumulate.
- If you accept the $20000, the amount you have after n months is the nth term of a geometric
sequence. How much will you have after 10 years?
- If you accept the $100 per month, the amount you have after n months is the nth partial sum of a
geometric series. How much will you have after 10 years?
- How long would it be before the amount you would have from the $100 per month
plan would exceed the amount from the lump sum plan?
- Show that if you can get 7% interest compounded monthly, the $100 per month
plan will never give you as much as the lump sum plan!
The IRS (Internal Revenue Service) assumes that the value of an item which can wear out decreases
by a constant number of dollars each year. For example, a house "depreciates" by 1/40 of its original
value each year.
What is your house worth after 1, 2, 3 years?
Do these values form an arithmetic or geometric sequence?
Why do you suppose the IRS calls this model "straight-line" depreciation?
Magic Squares:
Arrange the numbers 1-9 into a square subdivided
into nine smaller squares so that the sum of every
row, column, and main diagonal is the same.
The result is called a magic square.
LOGIC
Logic is a tool used in mathematical thinking and problem solving. It is essential for reasoning. In
logic, a statement is a sentence that is either true or false, but not both.
ex: 2 + 3 = 5
A hexagon has 6 sides.
These are not statements:
x + 3 = 5
She has blue eyes.
From a given statement, it is possible to create a new statement by forming a negation. The negation
of a statement is a statement with the opposite truth value of the given statement.
ex: It is snowing.
Negation: It is not snowing.
There is a symbolic system defined to help in the study of logic. If p represents a statement, the
negation of the statement p is denoted by ~p and is read "not p." Truth tables are used to show all
possible true-false values for statements.
ex:
statement
p
negation
~p
T
F
F
T
Quantifiers: Some statements involve quantifiers and are more complicated to negate. Quantifiers
include words such as all, some, every, and there exists.
- All, every, and no refer to each and every element in a set.
- Some and there exists at least one refer to one or more, or possibly all of the elements in a set.
- All, every, and each have the same mathematical meaning as do some and there exists at least one.
ex: "Some students at U of I have blue eyes" means at least one or possibly all have blue eyes.
Negation: "No students at U of I have blue eyes."
ex: "All students like hamburgers."
Negation: "Some students do not like hamburgers."
From two given statements, it is possible to create a new, compound statement by using a connective
such as and or or.
Ex: It is snowing. The ski run is open.
It is snowing and the ski run is open.
This compound statement is denoted by p q or p V q.
The symbols and V are used to represent and & or.
"And" is called a conjunction , and "or" is called a disjunction.
Because each statement p and q may be either true or false, there are 4 distinct possibilities for the
truth tables.
p q p q p q p V q
T
T
F
F
T
F
T
F
T
F
F
F
T
T
F
F
T
F
T
F
T
T
T
F
Two statements are logically equivalent if, and only if, they have the same truth values. Truth tables
are used to determine if two statements are logically equivalent, written p q.
Statements expressed in the form "if p, then q" are called conditionals , or implications , and are
denoted p – > q. The "if" part is called the hypothesis, and the "then" part is called the conclusion.
Many statements can be put in "if-then" form.
ex: All equilateral triangles have acute angles.
If a triangle is equilateral, then it has an acute angle.
p q p – > q
T
T
F
F
T
F
T
F
T
F
T
T
To show that an argument is valid, all possible Venn diagrams must show that there are no
contradictions. There must be no way to satisfy the hypothesis and contradict the conclusion if the
argument is valid, i.e., to show an argument is not valid, you need only draw a picture that satisfies the
hypothesis and contradicts the conclusion.
ex: hypothesis: All school teachers are mathematically literate.
Some mathematically literate people are not children.
Conclusion: Therefore no school teacher is a child.
A different method for determining whether an argument is valid uses direct reasoning and a form of
argument called the law of detachment or modus ponens.
[(p – >q) p] – >q
ex: If the sun is shining, then we shall take a trip.
The sun is shining.
Using the two statements, we can conclude that we shall take a trip.
In general, if the statement "if p then q" is true, then q must be true.
Another type of reasoning, indirect reasoning, uses a form of argument called modus tollens. If the
conditional is true, and we know the conclusion is false, then the hypothesis must be false.
[(p – >q) ~q] – >p
ex: If a figure is a square, then it is a rectangle.
The figure is not a rectangle.
Conclusion: The figure cannot be a square.
Determine a conclusion for the true statements.
If x = 3, then 2x ≠ 7.
We know that 2x = 7.
Therefore,
The final reasoning argument to be considered involves the chain rule (also known as hypothetical
syllogism).
ex: If I save, I will retire early.
If I retire early, I will become lazy.
Therefore,
In general, "if p then q" and "if q then r" are true, then "if p then r" is true.
Consider the following argument:
If a number is a power of 3, then it ends in a 1, 3, 7, or 9.
The number 3124 does not end in a 1, 3, 7, or 9.
Therefore, 3124 is not a power of 3.
p q p – >q (p–>q) ~q ~p
T
T
F
F
T
F
T
F
T
F
T
T
F
F
F
T
F
F
T
T
This argument is an application of modus tollens.
Draw an Venn diagram to represent the following argument and decide whether it is valid.
All timid creatures (T) are bunnies (B).
All timid creatures are furry (F).
Some cows (C) are furry.
Therefore, all cows are timid creatures.
Which of the laws (modus ponens, chain rule, or modus tollens) is being used in each of the following
arguments?
- If Joe is a professor, then he is learned. If your are learned, then you went to college. Joe is a
professor, so he went to college.
- If you have children, then you are an adult. Bob is not an adult, so he has no children.
- If a number ends in zero, then it is a multiple of 10. Forty is a number that ends in zero, so it is a
multiple of 10.
