5.1.1
First, we define f-1 as f-1[f(s)]=s
As f is one to one and onto, which means that for every s in the set S, the function f maps s to only one
element f(s) in the set S and for every f(s) in the set S there is an element s in the set S that maps s to
7.2.1
Given two different points P and Q, suppose they lie on anywhere in the boundary circle except
the boundary. Then, we can construct a chord that goes through both P and Q within the circle.
Since this chord would be unique, postulate 1 is proven
Sec
161
HW2
2.1.1
Alternate interior angles:
From theorem 2.9 we know that DAB=FBG.
As ABE and FBG are vertical angles constructed by AG and EF, they are congruent.
Thus DAB=ABE
Q.E.D.
Sum of interior angles on the same side:
From theorem 2.9 we know that DAB
Zian Deng
20211966
3.5.2
As eix= cosx+isinx, we have ei=cos+isin=-1+0=-1
Thus ei+1=0
3.5.4
Suppose z= r(cos +i sin ), then 1/z=1/r*(cos -i sin ) satisfieds z*1/z=1 as r*1/r=1,
(cos +i sin )* (cos -i sin )= (cos 2+ sin2 )=1
Q.E.D.
1.
(a)When z1, z2a, h(z1)
Zian Deng
Deng 1
20211966
HW3
161
1.
(a)i.6
ii.4
iii.4
iv.4
(b)According to Lagranges Theorem, if (m) is the number of elements in the ring Zm, and em(a) is
the order of one of the elements, then em(a) divides (m)
3. (a)If k is even, then we have (g k/2 )
HW8
Zian Deng
7.5.2
For any quadrilateral in hyberbolic geometry, it can always be divided into two triangles. Using
Theorem 7.15 that every triangle in hyperbolic geometry has a angle sum of less than 180
degrees, we can know that for all quadrilaterals,
MATH 161 SAMPLE MIDTERM EXAM
2014 MAY 5
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden from view during the exam.
Cheating in any form
MATH 161 SAMPLE MIDTERM EXAM SOLUTIONS
2014 MAY 5
Problem 1 (3 points). State, but do not prove, the SAS (side-angle-side) similarity theorem.
Solution. For any two triangles ABC and A B C , if A A and AB/A B =
=
AC/A C , then ABC A B C .
Problem 2 (5 poi
MATH 161 SAMPLE FINAL EXAM
SPRING 2014
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden from view during the exam.
Cheating in any form
MATH 161 SAMPLE FINAL EXAM
SOLUTIONS
SPRING 2014
1. Euclidean geometry
In this section, you may use the parallel postulate and its consequences.
Problem 1 (4 points).
(1) State Playfairs postulate.
(2) Dene the reection across a line .
Solution.
(1) For e
MATH 161 MIDTERM EXAM
2014 MAY 5
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden from view during the exam.
Cheating in any form may re
MATH 161 MIDTERM EXAM
2014 MAY 5
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden from view during the exam.
Cheating in any form may re
Zian Deng
20211966
Math 161
HW1
1.4.2
Suppose the first pile has m coins, the second has n coins, m>n, then the first player can take m-n from
the first pile, and then no matter how many coins the second player takes from whichever pile, the first
player