HOMEWORK 4 SOLUTIONS TO SELECTED PROBLEMS
1.
Let
H
Chapter 3, Problem 18 (Graded)
K be subgroups of G. Then e, the identity, must be in H and K , so it
H K . Thus, H K is nonempty, so we can use either Theorem 3.1 or
and
must be in
Theorem 3.2 (the one-st
Jesus Parables in Chronological Order
Parable #37 Luke 19:12-27 Ten Servants and Minas
11
While they were listening to this, he went on to tell them a parable, because
he was near Jerusalem and the people thought that the kingdom of God was going
to appea
Jesus Parables in Chronological Order
Parable #35 Luke 18:1-17 The Persistent Widow
1
Then Jesus told his disciples a parable to show them that they should always
pray and not give up. 2 He said: In a certain town there was a judge who neither
feared God
Jesus Parables in Chronological Order
Parable #34 Matthew 20:1-16 Workers in the Vineyard
1
For the kingdom of heaven is like a landowner who went out early in the
morning to hire men to work in his vineyard. 2 He agreed to pay them a denarius
for the day
Jesus Parables in Chronological Order
Parable #33 Luke 16:19-31 The Rich Man and Lazarus
19
There was a rich man who was dressed in purple and fine linen and lived in
luxury every day. 20 At his gate was laid a beggar named Lazarus, covered with
sores 21
Jesus Parables in Chronological Order
Parable #32 Luke 16:1-8 The Shrewd Manager
1
Jesus told his disciples: There was a rich man whose manager was accused of
wasting his possessions. 2 So he called him in and asked him, What is this I hear
about you? Giv
Parable #31 Luke 15:11-32 The Prodigal Son
11
Jesus continued: There was a man who had two sons. 12 The younger one
said to his father, Father, give me my share of the estate. So he divided his
property between them.
13
Not long after that, the younger so
Jesus Parables in Chronological Order
Parable #29 Luke 15:3-7 The Lost Sheep
3
Then Jesus told them this parable: 4 Suppose one of you has a hundred sheep
and loses one of them. Does he not leave the ninety-nine in the open country and
go after the lost s
Jesus Parables in Chronological Order
Parable #28 Luke 14:25-35 The Cost of Being a Disciple
25
Large crowds were traveling with Jesus, and turning to them he said:
If anyone comes to me and does not hate his father and mother, his wife and
children, his
Parable #26 Luke 14:7-14 Lowest Seat at a Feast
7
When he noticed how the guests picked the places of honor at the table, he
told them this parable: 8 When someone invites you to a wedding feast, do not
take the place of honor, for a person more distingui
Jesus Parables in Chronological Order
Parable #24 Luke 10:25-42 The Good Samaritan
25
On one occasion an expert in the law stood up to test Jesus. Teacher, he
asked, what must I do to inherit eternal life?
26
What is written in the Law? he replied. How do
Jesus Parables in Chronological Order
Parable #22 Luke 17:7-10 Master and Servant
7
Suppose one of you had a servant plowing or looking after the sheep. Would
he say to the servant when he comes in from the field, Come along now and sit
down to eat? 8 Wou
Jesus Parables in Chronological Order
Parable #20 Matthew 18:12-14 Lost Sheep
12
What do you think? If a man owns a hundred sheep, and one of them
wanders away, will he not leave the ninety-nine on the hills and go to look for the
one that wandered off? 1
Jesus Parables in Chronological Order
Four Kingdom of Heaven Parables
Parable #16 Matthew 13:44 Hidden Treasure
44
The kingdom of heaven is like treasure hidden in a field. When a man found
it, he hid it again, and then in his joy went and sold all he had
Jesus Parable in Chronological Order
Three Kingdom of Heaven Parables
Parable #13 Mark 4:26-29 The Growing Seed
26
He also said, This is what the kingdom of God is like. A man scatters seed
on the ground. 27 Night and day, whether he sleeps or gets up, th
Jesus Parables in Chronological Order
Parable #12 Matthew 13:24-30, 36-43 Weeds Among Good Plants
24
Jesus told them another parable: The kingdom of heaven is like a man who
sowed good seed in his field. 25 But while everyone was sleeping, his enemy came
Parable #38 Matthew 21:28-32 The Two Sons
28
What do you think? There was a man who had two sons. He went to the first
and said, Son, go and work today in the vineyard.
29
I will not, he answered, but later he changed his mind and went.
30
Then the fathe
Jesus Parables in Chronological Order
Parable #40 Matthew 22:1-14 Wedding Banquet
1
Jesus spoke to them again in parables, saying: 2 The kingdom of heaven is
like a king who prepared a wedding banquet for his son. 3 He sent his servants to
those who had b
Jesus Parables in Chronological Order
Parable #41 Matthew 24:32-35 Signs From a Fig Tree
32
Now learn this lesson from the fig tree: As soon as its twigs get tender and
its leaves come out, you know that summer is near. 33 Even so, when you see all
these
HOMEWORK 3 SOLUTIONS TO SELECTED PROBLEMS
Chapter 2, Problem 10. First, be aware that the two subsets of
H
same.
is basically the square elements of
x D4 . K
D4 :
D4
elements which equal
are not the
x2
for some
is the set of elements whose square is the i
Math 110A HW 3.3 Solutions
11. Let H be a subgroup of G, let a be a xed element of G, and let
K = cfw_x G|x = aha1 for some h H .
Prove that K is a subgroup of G.
Proof. Assume H G and let a G be xed. K is a subset of G by denition.
Since H is a subgroup
HOMEWORK 2 SOLUTIONS TO SELECTED PROBLEMS
Chapter 0, Problem 18. To simplify
which is equal to either
1
or
1
8402 mod 5,
we rst look for a power of
8
mod 5.
82 mod 5 = 64 mod 5 = 1 mod 5
Thus, we have
8402 mod 5 = 82
201
mod
5 = (1)201 mod 5 = 1 mod 5 = 4
Chapter 0, Problem 20
Let p1 , . . . , pn be a collection of primes. We must show that pi does not divide
p1 . . . pn + 1.
Method 1: Proof by contradiction
Suppose pi divides p1 . . . pn + 1. Then by denition,
p1 . . . pn + 1 = mpi
for some integer m. Not
Math 481, Abstract Algebra, Winter 2004
Problem Set 3, due Friday, January 23
3.1 For each group in the following list, nd the order of the group and the order of each element
in the group. What relation do you see between the orders of the elements of a
1. Prove: If H and K are subgroups of a group G, then H K is a subgroup of G. Proof. Premises: H and K are subgroups of a group G First we must show that H K is a nonempty subset of G. Let x H K be PBAC. Thus H K G. H K is nonempty because eG H and eG K,
Selected Homework Solutions
14 December 2003
Chapter 3
3.6: We have that x2 = e and x6 = e.
By way of contradiction, assume that x4 = e. Then e = x6 x4 = x2 ,
which contradicts our assumption that x2 is not the identity. Therefore
x4 = e.
By way of contra
First Midterm: Answers
MA441: Algebraic Structures I
12 October 2003
1) Denitions
1. Dene what it means for a subset of a group to be a subgroup.
A subset H of G that is itself a group under the operation of G is a
subgroup.
2. Dene what it means for a gr
Woundedness and Hope for Faith Today
From an article published in Italian in StudiaPatavina 51
(2004), 613-630.
THREE CULTURAL WOUNDS
Karl Rahner stressed that the crucial change affecting
faith in the modern era lies in the conditions surrounding
our fre
The Bible as Literature
Author(s): J. E. McFadyen
Reviewed work(s):
Source: The Biblical World, Vol. 16, No. 6 (Dec., 1900), pp. 438-446
Published by: The University of Chicago Press
Stable URL: http:/www.jstor.org/stable/3136951 .
Accessed: 08/11/2012 03
Christian Higher Education, 5:8396
Copyright C 2006 Taylor & Francis Inc.
ISSN: 1536-3759 print / 1539-4107 online
DOI: 10.1080/15363750500382774
THEOLOGY AND IMAGINATION: FROM THEORY
TO PRACTICE
MICHAEL PAUL GALLAGHER, S.J.
Gregorian University, Rome, It