MATH 310 Homework due 07/26/2013 1. Show that the vectors (1/(3 √ 2), 1/(3 √ 2),−4/(3 √ 2))T , (2/3, 2/3, 1/3)T , (1/ √ 2,−1/ √ 2, 0)T form an orthonormal basis of R3. Find the coordinates of the vector (1, 4, 3)T with respect to that basis. 2. Find an orthonormal basis of the image of the linear map: T : R2 → R3 with T((x, y)T ) = (3x−2y, x + y, x−y)T 3. Apply the Gramm-Schmidt orthogonalization process to the vectors (1, 3, 2)T and (1, 0, 1)T in order to get an orthonormal basis of the subspace that they span. 4. Find an orthonormal basis of the kernel of the linear map T : R3 → R with T((x, y, z)T ) = x−3y + z. 5. Find an orthonormal basis of the subspace: V = {(x, y, z, w)T : x + y + z + w = 0} of R4. 1 Logic A Concise Introduction to Logic by Hurley Available online if instructions need to be clarified. Tenth Edition Put the following syllogisms into standard form using letters to represent the terms, and name the mood and figure. Then use the two lists of valid syllogistic forms to determine whether each is valid from the Boolean standpoint, valid from the Aristotelian standpoint, or invalid. (need 6,8,9 completed-7 and 10 example answers) 6. Some African American nations are countries deserving military aid, because some African American nations are not upholders of human rights, and all countries deserving military air are upholders of human rights. 7.All pranksters are exasperating individuals, consequently some leprechauns are exasperating individuals, since all leprechauns are pranksters. All P are E All L are P Some L are E AAI- invalid 8. Some racists are not people suited to be immigration officials, given that some humanitarians are not people suited to be immigration officials, and no humanitarians are racists. 9. No people who respect human life are terrorists, and all airline hijackers are terrorists. Hence, no airline hijackers are people who respect human life. 10. Some silicates are crystalline substances, because all silicates are oxygen compounds, and some oxygen compounds are not crystalline substances. III. Reconstruct the syllogistic forms from the following combinations of mood and Some O are not C All S are O Some S are O OAI-1 invalid Reconstruct the syllogistic forms from the following combinations of moods and figure. NEED 7,8,9 6. EAO-4 E No M are P A All M are S O Some S are not P 7. AAA-1 A A A 8. EAO-2 E A O 9. OEI-3 O E I 10. OEA-4 O Some P are not M E No M are S A All S are P For the following- Use Venn diagrams to determine whether the following standard form categorical syllogism are valid from the Boolean standpoint, valid from the Aristotelian standpoint, or invalid. The identify the mood and figure. (16 and 19 example) Need 15,17, 20 15. Some ...

1. MATH 310
Homework due 07/26/2013
1. Show that the vectors
(1/(3
√
2), 1/(3
√
2),−4/(3
√
2))T , (2/3, 2/3, 1/3)T , (1/
√
2,−1/
√
2, 0)T
form an orthonormal basis of R3. Find the coordinates of the
vector
(1, 4, 3)T with respect to that basis.
2. Find an orthonormal basis of the image of the linear map: T :
R2 → R3
with
T((x, y)T ) = (3x−2y, x + y, x−y)T
3. Apply the Gramm-Schmidt orthogonalization process to the

2. vectors
(1, 3, 2)T and (1, 0, 1)T in order to get an orthonormal basis of
the
subspace that they span.
4. Find an orthonormal basis of the kernel of the linear map T :
R3 → R
with T((x, y, z)T ) = x−3y + z.
5. Find an orthonormal basis of the subspace:
V = {(x, y, z, w)T : x + y + z + w = 0}
of R4.
1
Logic
A Concise Introduction to Logic by Hurley
Available online if instructions need to be clarified.
Tenth Edition
Put the following syllogisms into standard form using letters to
represent the terms, and name the mood and figure. Then use
the two lists of valid syllogistic forms to determine whether
each is valid from the Boolean standpoint, valid from the
Aristotelian standpoint, or invalid. (need 6,8,9 completed-7 and
10 example answers)
6. Some African American nations are countries deserving
military aid, because some African American nations are not
upholders of human rights, and all countries deserving military
air are upholders of human rights.

3. 7.All pranksters are exasperating individuals, consequently
some leprechauns
are exasperating individuals, since all leprechauns are
pranksters.
All P are E
All L are P
Some L are E
AAI-
invalid
8. Some racists are not people suited to be immigration
officials, given that
some humanitarians are not people suited to be immigration
officials, and no
humanitarians are racists.
9. No people who respect human life are terrorists, and all
airline hijackers
are terrorists. Hence, no airline hijackers are people who respect
human
life.
10. Some silicates are crystalline substances, because all
silicates are oxygen
compounds, and some oxygen compounds are not crystalline
substances.
III. Reconstruct the syllogistic forms from the following
combinations of mood and

4. Some O are not C
All S are O
Some S are O
OAI-1
invalid
Reconstruct the syllogistic forms from the following
combinations of moods and figure. NEED 7,8,9
6. EAO-4
E No M are P
A All M are S
O Some S are not P
7. AAA-1
A
A
A
8. EAO-2
E
A
O
9. OEI-3

5. O
E
I
10. OEA-4
O Some P are not M
E No M are S
A All S are P
For the following- Use Venn diagrams to determine whether the
following standard form categorical syllogism are valid from
the Boolean standpoint, valid from the Aristotelian standpoint,
or invalid. The identify the mood and figure. (16 and 19
example) Need 15,17, 20
15. Some communications satellites are rocket-launched
failures.
All communications satellites are devices with antennas.
Therefore, some devices with antennas are rocket-launched
failures.
16. All currently living dinosaurs are giant reptiles.
All giant reptiles are ectothermic animals.

6. Therefore, some ectothermic animals are currently living
dinosaurs.
All C are G
All G are E
Some E are C
AAI-4
invalid
17. All survivalists are people who enjoy simulated war games.
No people who enjoy simulated war games are soldiers who
have tasted the
agony of real war.
Therefore, all soldiers who have tasted the agony of real war are
survivalists.
18. No spurned lovers are Valentine’s Day fanatics.
Some moonstruck romantics are Valentine’s Day fanatics.
Therefore, some moonstruck romantics are not spurned lovers.

7. 19. No theocracies are regimes open to change.
All theocracies are governments that rule by force.
Therefore, some governments that rule by force are not regimes
open to
change.
No S are I
All S are N
Some N are not I
EAO-3
Invalid, Boolean,
Valid Aristotelian
20. Some snowflakes are not uniform solids.
All snowfl akes are six-pointed crystals.
Therefore, some six-pointed crystals are not uniform solids
6-10 NEED HELP WITH 6,7,9
Use a VENN diagram to obtain the conclusion that is validly
implied by each of the following sets f premises. If no

8. conclusion can be validly dram, “no conclusion”
6
. No M are P.
Some S are not M.
Conclusion
7
. All M are P.
All S are M.
Conclusion All S are P
8
. All P are M.
All S are M.
Conclusion All S are P
9
. No P are M.
Some M are S.
Conclusion No P are S
10. No P are M.
No M are S.
Conclusion : NONE

9. 15-20 NEED HELP WITH 15,17, 18, 20
Reconstruct the following syllogistic forms and use the five
rules for syllogisms to determine is they are valid from the
Boolean standpoint, conditionally valid from the Aristotelian
standpoint, or invalid. For those that are conditionally valid,
identify the condition that must be fulfilled.
15. EAO-3
16 EEE-1 No M are P
No S are M
No S are P
Invalid;
Exclusive premises
17 EAE -1

10. 18 OAI -3
19 AOO 2
All P are M
Some S are nt M
Some S are not P
Valid; Boo ean No rules broken
20 EAO 1