3110 Final Exam | ||

Login ID: drsarah |
Attempt: 1 / 1 |
Max. Score: 60 |

Quiz Started: May 08, 2000 8:46 |
Quiz Finished: May 08, 2000 9:00 |
Time Spent: 14 min., 35 sec. |

Student finished 1 hr., 15 min., 25 sec. ahead of the 90 min. time limit. |

y=cos(x) | y is 1-1 and onto | |

y=e^x | y is not a function | |

y=x^3-x | y is onto but not 1-1 | |

y=x | y is 1-1 but not onto | |

x=y^2 | y is neither 1-1 nor onto |

**Student Response:**

y=cos(x) | --> |
y is neither 1-1 nor onto
| Correct |

y=e^x | --> |
y is 1-1 but not onto
| Correct |

y=x^3-x | --> |
y is onto but not 1-1
| Correct |

y=x | --> |
y is 1-1 and onto
| Correct |

x=y^2 | --> |
y is not a function
| Correct |

**Score:** 9.0 / 9.0

x^5-2x^3-8x-2 | If at all reducible, then solvable by radicals | |

x^5 | always solvable by radicals | |

x^5-1 | partially reducible, solvable by radicals | |

5th degree polynomials | completely reducible, solvable by radicals | |

4th or lower degree polynomials | irreducible, not solvable by radicals |

**Student Response:**

x^5-2x^3-8x-2 | --> |
irreducible, not solvable by radicals
| Correct |

x^5 | --> |
completely reducible, solvable by radicals
| Correct |

x^5-1 | --> |
partially reducible, solvable by radicals
| Correct |

5th degree polynomials | --> |
If at all reducible, then solvable by radicals
| Correct |

4th or lower degree polynomials | --> |
always solvable by radicals
| Correct |

**Score:** 9.0 / 9.0

2Z | integral domain, not a field | |

quaternions | integral domain and field | |

unit quaternions | ring which is abelian under *, has a mult id, but is not an integral domain | |

symetries of tetrahedron | ring which is abelian under *, has no zero divisors, but is not an integral domain | |

Z_6 | non-abelian group, not a ring | |

Z | ring which has no zero divisors, has a mult id, but is not an integral domain | |

Z_p, where p is prime | abelian group, not a ring |

**Student Response:**

2Z | --> |
ring which is abelian under *, has no zero divisors, but is not an integral domain
| Correct |

quaternions | --> |
ring which has no zero divisors, has a mult id, but is not an integral domain
| Correct |

unit quaternions | --> |
non-abelian group, not a ring
| Correct |

symetries of tetrahedron | --> |
abelian group, not a ring
| Correct |

Z_6 | --> |
ring which is abelian under *, has a mult id, but is not an integral domain
| Correct |

Z | --> |
integral domain, not a field
| Correct |

Z_p, where p is prime | --> |
integral domain and field
| Correct |

**Score:** 9.0 / 9.0

Match the statements on the left with the **incomplete proof** on the right that (when completed)
**proves or disproves** the statement.

a,b,c in R, a not 0, ab=ac implies b=c | Let R=Z_6={0,1,2,3,4,5}. Take a=0, b=1. Notice that a,b in R, ab=0*1=0 and a=0, as desired. | |

a,b, in R, ab=0 implies a=0 or b=0 | Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=4. Notice that a,b,c in R and a not 0. Also, ab=2*1=2=2mod6=8mod6=2*4=ac, but 1 is not equal to 4, as desired. | |

a in R implies, a not 0, a^2 = a implies a=0 or a=1. | Let R=Z_6={0,1,2,3,4,5}. Take a=4. Notice a in R, a not 0, a^2=4^2=16=4mod6 and a not 0 and a not 1, as desired. | |

There exists R, there exists a,b, in R s.t. ab=0 and (a=0 or b=0) | Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=1. Notice a,b,c in R, a not 0, ab=2*1=2=2*1=ac and b=1=c, as desired. | |

There exists R, there exists a in R s.t. a not 0, a^2=a and (a=0 or a=1) | Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=3. Notice that a,b in R and ab=2*3=6=0mod6 and 2 is not 0 and 3 is not 0, as desired. | |

There exists R, there exists a,b,c in R s.t. a not 0, ab=ac and b=c. | Let R=Z_6={0,1,2,3,4,5}. Take a=1. Notice a in R, a not 0, a^2=1^2=1, and a=1, as desired. |

**Student Response:**

a,b,c in R, a not 0, ab=ac implies b=c | --> |
Let R=Z_6={0,1,2,3,4,5}.
Take a=2, b=1, c=4. Notice that a,b,c in R and a not 0.
Also, ab=2*1=2=2mod6=8mod6=2*4=ac, but 1 is not equal to 4, as desired.
| Correct |

a,b, in R, ab=0 implies a=0 or b=0 | --> |
Let R=Z_6={0,1,2,3,4,5}.
Take a=2, b=3. Notice that a,b in R and ab=2*3=6=0mod6 and
2 is not 0 and 3 is not 0, as desired.
| Correct |

a in R implies, a not 0, a^2 = a implies a=0 or a=1. | --> |
Let R=Z_6={0,1,2,3,4,5}.
Take a=4. Notice a in R, a not 0, a^2=4^2=16=4mod6 and a not 0 and a not 1, as desired.
| Correct |

There exists R, there exists a,b, in R s.t. ab=0 and (a=0 or b=0) | --> |
Let R=Z_6={0,1,2,3,4,5}.
Take a=0, b=1. Notice that a,b in R, ab=0*1=0 and a=0, as desired.
| Correct |

There exists R, there exists a in R s.t. a not 0, a^2=a and (a=0 or a=1) | --> |
Let R=Z_6={0,1,2,3,4,5}.
Take a=1. Notice a in R, a not 0, a^2=1^2=1, and a=1, as desired.
| Correct |

There exists R, there exists a,b,c in R s.t. a not 0, ab=ac and b=c. | --> |
Let R=Z_6={0,1,2,3,4,5}. Take a=2, b=1, c=1. Notice a,b,c in R, a not 0, ab=2*1=2=2*1=ac and b=1=c, as desired.
| Correct |

**Score:** 14.0 / 14.0

FOLLOWED BY a Pi rotation about the line connecting the midpoints of AB and CD of the tetrahedron

A

D B

C

sends the vertices to (see one of the model tetrahedrons):

0.0% |
1. | A D B C | |

0.0% |
2. | A C D B | |

0.0% |
3. | A B C D | |

0.0% |
4. | C A B D | |

0.0% |
5. | D C B A | |

100.0% |
6. | D B A C | |

0.0% |
7. | B A D C | |

0.0% |
8. | B D C A | |

0.0% |
9. | C D A B | |

0.0% |
10. | D A C B | |

0.0% |
11. | C B D A | |

0.0% |
12. | B C A D |

**Score:** 5.0 / 5.0

1. Niels Abel | a. proved that x^5+y^5=z^5 has no non-zero whole number solutions. | |

2. Marjorie Lee Browne | b. The quaternions are a ring that are not abelian under multiplication since ij=k, but ji=-k | |

3. Julius Wihelm Richard Dedekind | c. When we adjoin
any root r of
f(x)=x^6+6x^5+17x^4+32x^3+37x^2+26x+6
to the complex numbers C,
we still get C. I.e.
C(r)=C for all roots r. | |

4. Leonhard Euler | d. Proved the Epsilon Conjecture which says that Taniyama-Shimura implies Fermat's Last Theorem | |

5. Pierre de Fermat | e. mentions the equation 8x8x8x8=4 in the cut "As" from Songs in the Key of Life. | |

6. Lodovico Ferrari | f. Z_3 is a finite field | |

7. Evariste Galois | g. proved that x^3+y^3=z^3 has no non-zero whole number solutions. | |

8. Johann Carl Friedrich Gauss (Fundamental Theorem of Algebra) | h. Z_6 is a ring but not a field
| |

9. Sophie Germain | i. Z_6={0,1,2,...,5} under +mod6 is the direct product of Z_2={0,1} under +mod2 and Z_3={0,1,2} under +mod3. | |

10. Sir William Rowan Hamilton
| j. Proved Fermat's Last Theorem
by proving Taniyama-Shimura | |

11. Leopold Kronecker (Fundamental Theorem of Finite Abelian Groups) | k. proved that x^4+y^4=z^4 has no non-zero whole number solutions. | |

12. Adrien-Marie Legendre | l. the circle is a manifold and a group under complex multiplication of norm 1 | |

13. Lie | m. Worked on the solution of a quartic by radicals | |

14. Francisco Maurolycus | n. Worked on the solution of a quintic by radicals | |

15. Emmy Amalie Noether | o. Z satisfies the ACC condition on ideals since every ascending chain of ideals terminates at some point. | |

16. Ken Ribet | p. First mathematician to examine a general approach (for all powers) for Fermat's Last Theorem | |

17. Peter Ludwig Mejdell Sylow (Sylow's First Theorem)
| q. the first mathematician known to use induction (in the proof that for all n in N, 1+3+5+...+(2n-1)=n^2. | |

18. Andrew Wiles | r. Z_6={0,1,2,...,5} under +mod6 has a subgroup of order 2 Z_2={0,3} under +mod6 and a subgroup of order 3 Z_3={0,2,4} under +mod6. | |

19. Stevie Wonder | s. 2x2 matrices with determinant 1 satisfying A times A transpose equals the identity form a group under matrix multiplication. |

**Student Response:**

1 | --> |
n | Correct |

2 | --> |
s | Correct |

3 | --> |
h | Correct |

4 | --> |
g | Correct |

5 | --> |
k | Correct |

6 | --> |
m | Correct |

7 | --> |
f | Correct |

8 | --> |
c | Correct |

9 | --> |
p | Correct |

10 | --> |
b | Correct |

11 | --> |
i | Correct |

12 | --> |
a | Correct |

13 | --> |
l | Correct |

14 | --> |
q | Correct |

15 | --> |
o | Correct |

16 | --> |
d | Correct |

17 | --> |
r | Correct |

18 | --> |
j | Correct |

19 | --> |
e | Correct |

**Score:** 100%