proposition involving the propositional
variables p, q, r, and s that is true when
exactly three of these propositional
variables are true and is false otherwise.
8. Show that these statements are
inconsistent: If Sergei takes the job offer
then he will g

integer. Find the truth values of 100
i=1(pi pi+1) and 100 i=1(pi pi+1).
19. Model 16 16 Sudoku puzzles (with
4 4 blocks) as satisfiability problems.
20. LetP (x) be the statement Student x
knows calculus and let Q(y) be the
statement Class y contains a s

using quantifiers, without using the
uniqueness quantifier. 36. Describe a rule
of inference that can be used to prove that
there are exactly two elements x and y in
a domain such that P (x) and P (y) are
true. Express this rule of inference as a
statemen

d) cfw_,cfw_,cfw_,cfw_ 21. Find the power
set of each of these sets, where a and b are
distinct elements. a) cfw_a b) cfw_a, b c) cfw_,
cfw_ 22. Can you conclude that A = B if A
and B are two sets with the same power
set? 23. How many elements does each o

= 0 uniqueness proof: a proof that there is
exactly one element satisfying a specified
property P1: 1/1 P2: 1/2 QC: 1/1 T1: 2
CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 Supplementary
Exercises 111 RESULTS The logical
equivalences given in

the final one conclusion: the final
statement in an argument or argument
form valid argument form: a sequence of
compound propositions involving
propositional variables where the truth of
all the premises implies the truth of the
conclusion valid argument

meant by a direct proof, a proof by
contraposition, and a proof by
contradiction of a conditional statement p
q. b) Give a direct proof, a proof by
contraposition and a proof by
contradiction of the statement: If n is
even, then n + 4 is even. 11. a) Des

Assuming the truth of the theorem that
states that n is irrational whenever n is a
positive integer that is not a perfect
square, prove that 2 + 3 is irrational.
Computer Projects Write programs with
the specified input and output. 1. Given
the truth valu

computational program or programs you
have written to do these exercises. 1. Look
for positive integers that are not the sum
of the cubes of nine different positive
integers. 2. Look for positive integers
greater than 79 that are not the sum of
the fourth

B B. 33. Find A2 if a) A = cfw_0, 1, 3. b) A =
cfw_1, 2, a, b. 34. Find A3 if a) A = cfw_a. b) A =
cfw_0, a. 35. How many different elements
does A B have if A has m elements and B
has n elements? 36. How many different
elements does A B C have if A has m

implies q): the proposition if p, then q,
which is false if and only if p is true and q
is false converse of p q: the conditional
statement q p contrapositive of pq:
the conditional statement q p inverse
of p q: the conditional statement p
q p q (bicondi

= y z(z = x) (z = y) such that this
statement is true. P1: 1/1 P2: 1/2 QC: 1/1
T1: 2 CH01-7T Rosen-2311T MHIA017Rosen-v5.cls May 13, 2011 15:27
Supplementary Exercises 113 23. Find a
domain for the quantifiers in xy(x = y
z(z = x) (z = y) such that this

bed; he considered these times his most
productive hours for thinking. Descartes
left school in 1612, moving to Paris,
where he spent 2 years studying
mathematics. He earned a law degree in
1616 from the University of Poitiers. At 18
Descartes became disg

that proceeds by showing that q must be
true when p is true proof by
contraposition: a proof that p q is true
that proceeds by showing that p must be
false when q is false proof by
contradiction: a proof that p is true based
on the truth of the conditiona

to have at least two subsets, the empty set
and the set S itself, that is, S and S
S. THEOREM 1 For every set S, (i) S
and (ii) S S. Proof: We will prove (i) and
leave the proof of (ii) as an exercise. Let S
be a set. To show that S, we must
show that x(

1
Administration Ethics
607/624
Hall Six
Truth in Advertising
and Environmental Stewardship
2
Week Six Topics
Marketing and Advertising
*
Three Articles
Two Cases
Environmental Stewardship
Two Articles
Three Cases
3
Marketing and Advertising
Market

1
Administration Ethics
607/624
Hall Five
Managing Human Resources and Finances
Ethically
2
Week Five Topics
Human Resources Management
*
Three Articles
Three Cases
Accounting and Finance
One Article
Three Cases
3
Human Resource Management
Question

1
Administration Ethics
607/624
Hall Three
Corporate Social Responsibility
from a Christian perspective
2
Reviewing Hall 2 issues-Christian Ethics for Life:
Norms and Benchmarks
Dual Ethics? Business
Ethics versus Personal
moral integrity?
Are the stand

Administration Ethics
607/624
Hall One
Understanding Ethics from a Christian Worldview
2
Topics well cover in 607/624
Review from 601
Belhavens vision and
mission
Understanding a Christian
Worldview
Student Responsibilities
and Course Expectations
Lea

CULTURE RELATIVISM
1
Culture Relativism
Shawandra Jones
Belhaven University-Desoto
CULTURE RELATIVISM
2
Culture relativism expresses the beliefs, ethics and customs of another are only relative to
the culture or individual that is expressing that belief.

of all commitments is maintained
throughout the test. 10. Suppose that in a
three-round obligato game, the teacher
first gives the student the proposition p
q, then the proposition (p r) q, and
finally the proposition q. For which of the
eight possible s

ordered triples (a, b, c), where a A, b
B, and c C. Hence, A B C = cfw_(0, 1, 0),
(0, 1, 1), (0, 1, 2), (0, 2, 0), (0, 2, 1), (0, 2,
2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0), (1,
2, 1), (1, 2, 2). Remark: Note that
when A, B, and C are sets, (A B)

(3, 3). We will study relations and their
properties at length in Chapter 9. Using
Set Notation with Quantifiers Sometimes
we restrict the domain of a quantified
statement explicitly by making use of a
particular notation. For example, xS(P
(x) denotes th

show that each set is a subset of the other.
In other words, we can show that if A and
B are sets with A B and B A, then A =
B. That is, A = B if and only if x(x A x
B) and x(x B x A) or
equivalently if and only if x(x A x
B), which is what it means for

codomain, the range, and the assignment
of values to elements of the domain.
EXAMPLE 1 What are the domain,
codomain, and range of the function that
assigns grades to students described in
the first paragraph of the introduction of
this section? Solution:

normals (also known as spies). Knights
always tell the truth, knaves always lie,
and normals sometimes lie and
sometimes tell the truth. Detectives
questioned three inhabitants of the island
Amy, Brenda, and Claireas part of the
investigation of a crime.

usually used to group together elements
with common properties, there is nothing
that prevents a set from having seemingly
unrelated elements. For instance, cfw_a, 2,
Fred, New Jersey is the set containing the
four elements a, 2, Fred, and New Jersey.
So

from consideration.] 49. a) Draw each of
the five different tetrominoes, where a
tetromino is a polyomino consisting of
four squares. b) For each of the five
different tetrominoes, prove or disprove
that you can tile a standard checkerboard
using these te

which he is best known. He also made
fundamental contributions to philosophy.
In 1649 Descartes was invited by Queen
Christina to visit her court in Sweden to
tutor her in philosophy. Although he was
reluctant to live in what he called the
land of bears a

an important role in discrete
mathematics: N = cfw_0, 1, 2, 3,., the set of
natural numbers Z = cfw_., 2, 1, 0, 1, 2,.,
the set of integers Z+ = cfw_1, 2, 3,., the set
of positive integers Q = cfw_p/q | p Z, q
Z, and q = 0, the set of rational numbers
R,