... intended as an
antidote to “Abstract Mathematics, ” since concrete classical results were rap-
idly being swept out of the modern mathematical curriculum by a new wave
of abstract ideas popularly called ... Therefore if we express f(n) in the form
f(n) = A( n) a + B(n)
B
+ C(n)y ,
(1.13)
Library of Congress Cataloging-in-Publication Data
Graham, Ronald Lewis,
1935-
Concrete mathematics
: afoundation ... hard one to the easy one.
time
to
do
warmup
Let’s apply these ideas to a useful example. Consider the array
exercises 4 and 6.)
(Or to check out
al
al
al
a2
the Snickers bar
a2 al
a2 a2
languishing...
... never have been found by a brute-force computer
search!
The symbols ∀ ( for all”) and ∃ (“there exists”) are called quantifiers. A quantifier is
always followed by a variable (and perhaps an indication ... indication of what values that variable
can take on) and then a predicate that typically involves that variable. The predicate
may itself involve more quantifiers. Here are a couple examples of statements ... prove that some statement holds for all natural values of a variable.
For example, here is a classic formula:
46 Number Theory I
generated lots of amazing ideas. But this lecture is about one...
... universal idea. Taking a walk is a literal example, but
so is cooking from a recipe, executing acomputer program, evaluating a formula,
and recovering from substance abuse.
Abstractly, taking a step ... for at least one x 2 R.
All these sentences “quantify” how often the predicate is true. Specifically, an
assertion that a predicate is always true is called a universal quantification, and an
assertion ... get around the ambiguity of English, mathematicians have devised a spe-
cial language for talking about logical relationships. This language mostly uses
ordinary English words and phrases such as...
... that in ascience of mind” sense CS has always existed. the criteria
current in any culture forscience may change greatly, but there always has been and
always will be ascience which deals ... supplying a
formalization of language has led to the formation of dozens of mutually antagonistic
camps, whose basic conceptions are often couched in highly baroque mathematical
formalisms? Can it ... little too far. His eventual aim was to explain logic
in action using the same set of concepts as for biological adaptation and psychological
processes. Above all, he insisted that Kantian categories...
... ADBEC BADCE BADEC
BDACE BDAEC DABCE DABEC DBACE DBAEC
that is, all lists in which A, B, and D precede C and E. Since there are 3! ways to arrange A,
B, and D, and 2! ways to arrange C and E,bythe ... more algebraically.
4. In how many ways can you draw a first card and then a second card from a deck of 52
cards?
5. In how many ways can you draw two cards from a deck of 52 cards.
6. In how many ... CHAPTER 1. COUNTING
the particular labelling in which A, B, and D are labelled blue and C and E are labelled red.
Which lists correspond to this labelling? They are
ABDCE ABDEC ADBCE ADBEC BADCE...
... and at great speed. Data are any kind of information that can be codified in some
manner and input into the computer. Normally, we think of data as facts and numbers such as a
person’s name and ... placed into a
memory area known as Number. I have also shown that the result is going to be placed in a
memory area known as Answer. Both Number and Answer are known as program variables. A
variable ... such as a transistor, has electricity, it can be said to contain a 1;
if none, then a 0. This is the basis forcomputer operations. The actual instructions that make up a
program are all in binary,...
... can also be shown that for any
relation R on a set A, (R ∪ R
−1
)
∗
is the least equivalence relation containing
R.
2.1.9 Partial and Total Orders
A relation R on a set A is antisymmetric iff for ... 3.2.3
A ∧ B ⊃ C is an abbreviation for ( (A ∧ B) ⊃ C),
A ∨ B ∧ C an abbreviation for (A ∨ (B ∧ C)), and
A ∨ B ∨ C is an abbreviation for ( (A ∨ B) ∨ C).
TABLE OF CONTENTS xix
6.5 Craig’s Interpolation ... the
following are tautologies.
A ∨ B ≡ A ∧ B ⊕ A ⊕ B
A ≡ A ⊕ 1
A ⊕ 0 ≡ A
A ⊕ A ≡ 0
A ∧ 1 ≡ A
A ∧ A ≡ A
A ∧ (B ⊕ C) ≡ A ∧ B ⊕ A ∧ C
A ∧ 0 ≡ 0
(iii) Prove that {⊕, ∧, 1} is functionally complete.
∗...
... boston, monday)
AA-57 departs from Boston at 8:00am.
equal (dtlme (aa-5T. boston), 8:00am)
AA-57 departs from Boston after 8:00am.
greater (dtime (aa-5T, boston), 8:00am)
A. A-57 departs from ... rule: base phrase structure rules and
transformational rules. It is also able to parse un-
grammatical sentences; it always uses the rule that
matches best, even if none match exactly. Paragram ... bound to a v~iable in a frame determiner, a
unique new name is generated for that variable and its bindings.
In this paper, we shall assume for simplicity that vaxiable names
~re maKically ~correct"...
... Cataloguing in Publication Data
Data available
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Data available
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With offices in
Argentina Austria Brazil Chile Czech Republic France Greece
Guatemala Hungary Italy Japan Poland ... The second part explained that forces could cause accelerations.
A question was left unanswered, ‘What causes the forces?’. You may have also found
it strange that one vital concept was completely...