Chapter 2
Macromechanical Analysis of a Lamina
Stress, Strain
Stressstrain relationships for different types of materials
Stressstrain relationships for a unidirectional/bidirectional lamina
Islamic Azad University, Najafabad Branch
53
Macromechanical Ana

integers. We are assuming that n(P (n)
Q(n) is true. Note that P (100) is true
because 100 > 4. It follows by universal
modus ponens that Q(100) is true, namely
that 1002 < 2100. Another useful
combination of a rule of inference from
propositional logic

reasoning is called the fallacy of denying
the hypothesis. EXAMPLE 11 Let p and q
be as in Example 10. If the conditional
statement p q is true, and p is true, is
it correct to conclude that q is true? In
other words, is it correct to assume that
you did

ocean pollution. c) Each of the 93
students in this class owns a personal
computer. Everyone who owns a personal
computer can use a word processing
program. Therefore, Zeke, a student in
this class, can use a word processing
program. d) Everyone in New Je

of 12 of these employees and check
whether they have the desired skills.
These examples show that it is often
necessary to generate permutations
and combinations to solve problems.
Generating Permutations Any set with
n elements can be placed in one-to-on

plays four games against each of the
other teams in this division, three
games against each of the 11
remaining teams in the conference,
and two games against each of the 16
teams in the other conference. In how
many different orders can the games of
one

Louvre. Therefore, someone in this class
has visited the Louvre. 15. For each of
these arguments determine whether the
argument is correct or incorrect and
explain why. a) All students in this class
understand logic. Xavier is a student in
this class. The

Rosen-2311T MHIA017-Rosen-v5.cls May
13, 2011 15:27 76 1 / The Foundations:
Logic and Proofs TABLE 2 Rules of
Inference for Quantified Statements. Rule
of Inference Name xP (x) P (c)
Universal instantiation P (c) for an
arbitrary c xP (x) Universal
genera

Permutations and Combinations 437
Generating Combinations How can we
generate all the combinations of the
elements of a finite set? Because a
combination is just a subset, we can
use the correspondence between
subsets ofcfw_a1, a2,.,an and bit strings
of

minus the number of zeros in the string b)
the function that assigns to each bit string
twice the number of zeros in that string c)
the function that assigns the number of
bits left over when a bit string is split into
bytes (which are blocks of 8 bits) d

of the elements of a set r-permutation:
an ordered arrangement of r elements
of a set P(n,r): the number of rpermutations of a set with n elements
r-combination: an unordered selection
of r elements of a set C(n,r): the
number of r-combinations of a set w

and 0 < 1. EXAMPLE 29 Prove that if x is
a real number, then 2x=x+x + 1 2 .
Solution: To prove this statement we let x
= n + , where n is an integer and 0 < 1.
There are two cases to consider,
depending on whether is less than, or
greater than or equal to

allows us to conclude that there is an
element c in the domain for which P (c) is
true if we know that xP (x) is true. We
cannot select an arbitrary value of c here,
but rather it must be a c for which P (c) is
true. Usually we have no knowledge of
what c

argument that establishes the truth of a
mathematical statement. A proof can use
the hypotheses of the theorem, if any,
axioms assumed to be true, and
previously proven P1: 1/1 P2: 1/2 QC:
1/1 T1: 2 CH01-7T Rosen-2311T
MHIA017-Rosen-v5.cls May 13, 2011
15

seven or more characters can be
formed from the letters in
EVERGREEN? 36. How many different
bit strings can be formed using six 1s
and eight 0s? 37. A student has three
mangos, two papayas, and two kiwi
fruits. If the student eats one piece of
fruit each

these functions from the set of teachers in
a school. Under what conditions is the
function one-to-one if it assigns to a
teacher his or her a) office. b) assigned
bus to chaperone in a group of buses
taking students on a field trip. c) salary. d)
social

Examples 12 and 13 we used both
universal instantiation, a rule of inference
for quantified statements, and modus
ponens, a rule of inference for
propositional logic. We will often need to
use this combination of rules of inference.
Because universal inst

homework, then I can answer all the
exercises. If I answer all the exercises, I
will understand the material. Therefore, if
I work all night on this homework, then I
will understand the material. 5. Use rules
of inference to show that the hypotheses
Randy

partial function fromZto R where the
domain of definition is the set of
nonnegative integers. Note that f is
undefined for negative integers.
Exercises 1. Why is f not a function from R
to R if a) f (x) = 1/x? b) f (x) = x? c) f (x)
= (x2 + 1)? 2. Determ

have six legs. Spiders eat dragon- flies.
d) Every student has an Internet
account. Homer does not have an
Internet account. Maggie has an Internet
account. e) All foods that are healthy to
eat do not taste good. Tofu is healthy to
eat. You only eat what

text. These include polynomial,
logarithmic, and exponential functions. A
brief review of the properties of these
functions needed in this text is given in
Appendix 2. In this book the notation log
x will be used to denote the logarithm to
the base 2 of x

for all elements in the domain of this
function. For example, a program may not
produce a correct value because
evaluating the function may lead to an
infinite loop or an overflow. Similarly, in
abstract mathematics, we often want to
discuss functions tha

b) If I eat spicy foods, then I have strange
dreams. I have strange dreams if there is
thunder while I sleep. I did not have
strange dreams. c) I am either clever or
lucky. I am not lucky. If I am lucky, then
I will win the lottery. d) Every computer
scie

proof can include axioms (or postulates),
which are statements we assume to be
true (for example, the axioms for the real
numbers, given in Appendix 1, and the
axioms of plane geometry), the premises,
if any, of the theorem, and previously
proven theorems

asymptotic to. Stirlings formula is named
after James Stirling, a Scottish
mathematician of the eighteenth century.
JAMES STIRLING (16921770) James
Stirling was born near the town of
Stirling, Scotland. His family strongly
supported the Jacobite cause of

then he is not a spider. George is a spider.
George has eight legs. 3. What rule of
inference is used in each of these
arguments? a) Alice is a mathematics
major. Therefore, Alice is either a
mathematics major or a computer science
major. b) Jerry is a m

happy. Given the premise xH (x), we
conclude that H(Lola). Therefore, Lola is
happy. 18. What is wrong with this
argument? Let S(x, y) be x is shorter than
y. Given the premise sS(s, Max), it
follows that S(Max, Max). Then by
existential generalization it

Many times conjectures are shown to be
false, so they are not theorems.
Understanding How Theorems Are Stated
Before we introduce methods for proving
theorems, we need to understand how
many mathematical theorems are stated.
Many theorems assert that a pr

boxes so that each of the boxes
contains at least one object? 56. How
many ways are there to pack eight
identical DVDs into five
indistinguishable boxes so that each
box contains at least one DVD? 57. How
many ways are there to pack nine
identical DVDs in