38
APPENDIX: MATHEMATICS REVIEW G
12.1.4
Eigenanalysis and Symmetric Matrices
Eigenvalues, Eigenvectors, and Symmetric Matrices play a fundamental role in describing
properties of spectral data obtained from imaging spectrometers. The definitions given
he
LINEAR ALGEBRA
39
Chapter 12 Exercises. Linear Algebra.
1. Let A and B be the following matrices:
1
A= 3
4
2
5
2
2 and B =
6 4
6
Compute the product C = AB.
2. Determine mathematically whether the set of vectors:
2
2
1
3 , 3
2 ,
B=
1
2
2
forms a
30
APPENDIX: MATHEMATICS REVIEW G
12.1.1
Matrices and Vectors
Definition of Matrix. An M xN matrix A is a two-dimensional array of numbers
2
6
6
A=6
4
a11
a21
a12
a22
.
.
.
.
.
.
.
aM 1
aM 2
.
.
a1N 3
a2N 7
7
. 7
. 5
aM N
A matrix can also be written as A
Math for Intelligent Systems - Fall 2017 - Approximate Calendar
September 7, 2017
Week
1
2
21
Norms, Metrics, and
Dissimilarities
3
28
SPDM
Diagonalization
4
4 Holiday Labor Day
5
11
20
Chapter 1
The Foundations: Logic and Proofs
d) We do not have to rule out your having more than one perfect friend. Thus we have simply :Jx(F(x)/\P(x).
Note the use of conjunction with existential quantifiers.
e) The expression is Vx(F(x) /\ P(x). Note
Section 1.3
Propositional Equivalences
15
35. We apply the rules stated in the preamble.
a) pV-.qV-.r
b) (pVqVr)/\s
c) (p/\T)V(q/\F)
37. If we apply the operation for forming the dual twice to a proposition, then every symbol returns to what it
originally
Section 1.4
Predicates and Quantifiers
17
SECTION 1.4 Predicates and Quantifiers
The reader may find quantifiers hard to understand at first. Predicate logic (the study of propositions with
quantifiers) is one level of abstraction higher than propositiona
Section 1.5
Nested Quantifiers
27
e) Vx\lyP(x,y), where P(x,y) is "x has been in y"; x ranges over students in this class, and y ranges over
buildings on campus
f) 3x3y\lz(P(z,y)-+ Q(x,z), where P(z,y) is 'z is in y" and Q(x,z) is "x has been in z"; x ran
Section 1.4
Predicates and Quantifiers
19
19. Existential quantifiers are like disjunctions, and universal quantifiers are like conjunctions. See Examples 11
and 16.
a) We want to assert that P(x) is true for some x in the universe, so either P(l) is true
Section 1.4
Predicates and Quantifiers
21
In English this reads "Every old dog is unable to learn new tricks" or "All old dogs can't learn new tricks."
(Note that this does not say that not all old dogs can learn new tricks-it is saying something stronger
28
Chapter 1
The Foundations: Logic and Proofs
d) This says that for every pair of real numbers x and y , there exists a real number z that is their sum. In
other words, the real numbers are closed under the operation of addition, another true fact. (Some
16
Chapter 1
The Foundations: Logic and Proofs
55. A truth table for a compound proposition involving p and q has four lines, one for each of the following
combinations of truth values for p and q: TT, TF, FT, and FF. Now each line of the truth table for
26
Chapter 1
The Foundations: Logic and Proofs
f) This is a little ambiguous in English. If the statement is that there is a very inquisitive student, one
who has gone around and asked a question of every professor, then this is similar to part ( d), with
22
Chapter 1
The Foundations: Logic and Proofs
43. A conditional statement is true if the hypothesis is false. Thus it is very easy for the second of these propositions
to be true-just have P(x) be something that is not always true, such as "The integer x
Section 1.5
Nested Quantifiers
23
53. a) This is certainly true: if there is a unique x satisfying P(x), then there certainly is an x satisfying P(x).
b) Unless the domain (universe of discourse) has fewer than two items in it, the truth of the hypothesis
Chapter 1
24
The Foundations: Logic and Proofs
1. a) For every real number x there exists a real number y such that x is less than y. Basically, this is asserting
that there is no largest real number-for any real number you care to name, there is a larger
Section 1.5
Nested Quantifiers
25
your school, there is some cuisine that at least one them does not like.
e) There are two students at your school who have exactly the same tastes (i.e., they like exactly the same
cuisines).
f) For every pair of students
Chapter 1
18
The Foundations: Logic and Proofs
the definition of the truth value of p -+ q). This illustrates the fact that you rarely want to use conditional
statements with existential quantifiers.
d) This statement is that there exists an x in the doma
Section 1.3
13
Propositional Equivalences
p
p
q
T
T
T
F
T
F
F
F
T
F
T
T
-'>
q
(p
-'>
q)
(p
-'>
F
q)
-'>
p
(p
q
-'>
q)
T
T
T
F
T
T
T
F
F
T
T
F
T
T
T
-'>
-,q
11. Here is one approach: Recall that the only way a conditional statement can be false is for the
Section 1.2
Applications of Propositional Logic
9
7. a) Since "whenever" means "if," we have q-+ p.
b) Since "but" means "and," we have q /\ 'P.
c) This sentence is saying the same thing as the sentence in part (a), so the answer is the same: q-+ p.
d) Ag
Section 1.3
Propositional Equivalences
11
37. If the first sign were true, then the second sign would also be true. In that case, we could not have one true
sign and one false sign. Rather, the second sign is true and the first is false; there is a lady i
Section 1.1
Propositional Logic
7
p
q
r
p _, q
p
'P _, r
(p _, q) V (p _, r)
(p _, q) A (p _, r)
T
T
T
T
F
F
T
T
T
T
T
F
F
F
F
F
T
F
T
F
T
F
T
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
T
T
F
F
F
T
F
F
F
T
F
T
F
For part ( e) we have
p
q
r
T
14
Chapter 1
The Foundations: Logic and Proofs
Therefore the entire hypothesis is false, so this assignment will not yield a false conditional statement. Since
we have argued that no assignment of truth values can make this proposition false, we have prov
Chapter 1
12
The Foundations: Logic and Proofs
3. We construct the following truth tables.
p
q
pVq
qVp
p/\ q
q /\p
T
T
T
T
T T
T
T
F
T F
F
F T
T
T
F
F
F
F
F
F F
F
Part (a) follows from the fact that the third and fourth columns are identical; part (b) fol
10
Chapter 1
The Foundations: Logic and Proofs
19. If A is a knight, then he is telling the truth, in which case B must be a knave. Since B said nothing, that
is certainly possible. If A is a knave, then he is lying, which means that his statement that at
5
Propositional Logic
Section 1.1
For part ( e) we have the following table. This time we have omitted the column explicitly showing the
negations of p and q. Note that this true proposition is telling us that a conditional statement and its
contrapositiv
The Foundations: Logic and Proofs
Chapter 1
6
35. The techniques are the same as in Exercises 31-34. For parts (a) and (b) we have the following table (column
four for part (a), column six for part (b).
p
q
q
T
T
F
F
T
F
T
F
F
T
F
T
p
-+
q
-.p
'P
F
F
T
T
2
Chapter 1
The Foundations: Logic and Proofs
e) Both parts of this biconditional statement are true, so by the truth-table definition this is a true statement.
9. This is pretty straightforward, using the normal words for the logical operators.
a) Sharks