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2.2 CONDITIONAL STATEMENTS
Connectives are used to join statements to create new statements: ->, ,
Definition
If p and q are statement variables,
p -> q :
p: the hypothesis or antecedent
q: the conclusion or consequent
the conditional of q by
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4.5 DIRECT PROOF AND COUNTEREXAMPLE V: FLOOR AND CEILING
Definition
Given any real number x, the floor of x, denoted that
x
= the unique integer n such that n x < n + 1
Symbolically, if x is a real number and n is an integer, then
x
=n
n x<n+1
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4.8 ALGORITHMS
Algorithms
A step by step method for performing some action (e.g., recipes, assembling directions)
Variable - A specific storage location
Assignment statement
variable := expression; / e.g., x = 3; x is assigned the value 3
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Chapter 4. Elementary Number Theory and Methods of Proof
Questions:
When the Floor of x is the largest integer that is less than or equal to x,
o For any real number x,
is Floor(x-1) = Floor(x) -1?
o For any real number x, and y,
is Floor(x-y)
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Chapter 2. The Logic of Compound Statements
2.1 LOGICAL FORM AND LOGICAL EQUIVALENCE
Statements
A statement (or proposition) is a sentence that is true or false but not both
Sentence Statement
e.g.
Two plus two equals four
T
Statement/Sentence
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4.4 DIVISION INTO CASES AND THE QUOTIENT-REMAINDER THEOREM
The Quotient Remainder Theorem
Given any integer n and positive integer d, there exist unique integers q and r such that
n = dq + r
where 0 r < d
_2
4 | 11
_8_
3
Quotient
(11 div 4)
Re
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2.3 VALID AND INVALID ARGUMENTS
Listed below are not the same. How are they different?
Sentences
Statements
Arguments
An argument is a sequence of statements
An argument form is a sequence of statement forms
Premises (or assumptions or hypothe
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4.6 CONTRADICTION AND CONTRAPOSITION
Proof by Contradiction
based on the fact that either a statement is true or false but not both
known as reductio ad impossibile, or reductio ad absurdum
(reducing a given assumption to an impossibility ab
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2.5 NUMBER SYSTEMS
1. Why different number systems?
There are only 2 states in a computer and these are represented by 0 and 1:
Binary Digits (bits)
The computer represents, stores, and manipulates all information in terms of Bits
2. Review
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4.2 DIRECT PROOF AND COUNTEREXAMPLE II: RATIONAL NUMBERS
o Definition
A real number r is Rational iff it can be expressed as a quotient of two integers with a
nonzero denominator. A real number that is not rational is Irrational
r is rational
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CH 3.4 ARGUMENTS WITH QUANTIFIED STATEMENTS
Valid Arguments with Quantifiers
Universal Instantiation
Universal Modus Ponens
Universal Modus Tollens
1. Universal Instantiation
o If some property is true of everything in a domain, then it is
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4.3 DIRECT PROOF AND COUNTEREXAMPLE III: DIVISIBILITY
Divisibility
If n and d are integers, then
n is divisible by d iff n = dk for some integer k
n is a multiple of d,
d is a factor of n,
d is a divisor of n,
d divides n
If n and d are intege
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CH 3.2 PREDICATES AND QUANTIFIED STATEMENTS II
Review
How can you show that a -quantified formula holds?
How can you show that a -quantified formula does NOT hold?
How can you show that a -quantified formula holds?
How can you show that a
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Chapter 3. The Logic of Quantified Statements
Compound Statements
Statements made of simple statements joined by the connectives ~, , , ->, and <->
Quantified Statements?
Questions:
1)
2)
If S is a man, then S is mortal
S is a man
Therefore, S
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4.7 THEOREMS
Note that every nonzero rational number has exactly one simplest form of ratio of two integers
with a positive denominator (e.g., 3/6 => 2/4 => 1/2). The simplest form is when the numerator
and the denominator have no common divis