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Some hints.

rumba(rumba)

2.1 Show nZ satisfy the four axioms under + operation, you can use the condition that (Z,+) is a group.
2.2 If m not exist, then S can not form a group.
3.1 Z is set of integer, (Z,+) form a group, identity is 0, inverse is negative.
Z4 is mod(Z,4) = {0,1,2,3}. (Z4,+) form a group, it preserves the identity (0) and operation (+) as group (Z,+), so f is homeomorphism. However, the inverse on (Z4, +) is 4-a if the original one is a.
3.2 Equivalent class of any integer x on Z of mod 4 operation is: sx = { ...., x-8, x-4, x, x+4, x+8, ....}.
To prove p is an equivalent relation, 1) Reflexive. sx = sx. 2) Symmetric. sx1 = sx2 ==> sx2 = sx1. 3) Transitive.
All distince equivalent class: {...x-4, x, x+4, ...} {...,x-5, x-1, x+3, ...} {...,x-6,x-2,x+2,...} {...,x-7,x-3,x+1,...}

(#1002712@0)
Last Updated: 2003-1-26
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工作学习 / 求学深造 / 离散数学，俺实在搞不定了:((( -rainrain(雨雨若我若无我); 2003-1-24 {937} (#998332@0)
1) Consider the structures ({a1x + a0 | a1, a0 属于 R}, +) and ({a + ib | a, b 属于R, i的平方 = -1}, +).

a) Show that they are groups
b) Show that they are isomorphic.

2) (Z, +) is a group. Denote by nZ the set of the multiple of n., i.e.
nZ = {a | a 属于Z and there exists c 属于Z s.t. a = nc}

a) Show that (nZ, +) is a subgroup of (Z, +).
b) Show that for any subgroup S of the additive group Z there is a natural number m such that S = mZ.

3) Let f : Z -> Z4 be defined by f(x) = x.4 1 (the non negative remainder when x is divided by 4) and Z4 = {0, 1, 2, 3}

a) Prove that f is a homeomorphism from the group (Z, +) onto the group (Z4, +4).To simulate (Z, +) by (Z4, +4) we use equivalence classes of Z. We define a relation p on Z by

xpy <->f(x) = f(y) <-> x.4 1 = y.4 1

b) Prove that p is an equivalence relation on Z (p is the relation of congruence modulo 4). Write all the distinct equivalence classes.

你们班上好像是你最牛的哟 -ppgg(聪明小子); 2003-1-24 (#998337@0)

和和，俺权当你是夸俺呢。这节课俺忘记带眼镜去了，这门课没有课本:( -rainrain(雨雨若我若无我); 2003-1-24 (#998345@0)

俺有中文的可以借你。:-) -becky(爱喝冷血的妖精); 2003-1-24 (#998347@0)

我的中文课本已经托家人海运过来了，快到了。到了之后，看几下，就能恢复从前读书的状态，可以帮rainrain了。 -ppgg(聪明小子); 2003-1-24 (#998364@0)

俺也有中文课本呀，可是...这怪俺自己，眼睛不是那么好还死活不愿意戴眼镜。 -rainrain(雨雨若我若无我); 2003-1-24 (#998384@0)

MM你要善于团结现实中一切可以团结的力量啊，不要把电话号码的保密等级设的太高啊。 -hardyfan(No Way Out); 2003-1-24 {45} (#998351@0)
不过我也知道这个这个。。。不是很好把握的。：)
此门课是唯一能把大雪时代精神百倍的我给活生生讲睡着的课, 当然得怨那个老师 -expertune(伪而不劣); 2003-1-24 (#998372@0)

我觉的离散挺有意思的啊 -rabbitbug(兔八哥); 2003-1-24 (#998392@0)

当时读的时候觉得有意思。可是后来因为一直没用，忘光啰 -ppgg(聪明小子); 2003-1-24 (#998425@0)

俺是学一次不会，再学一次还不会，估计学几百次还是不会的那种笨人，和和 -rainrain(雨雨若我若无我); 2003-1-24 (#998434@0)

离散讲的就是逻辑推理吧，虽说用的地方不大，其实我们都是潜移莫化中用到了。女孩子讲的是无理逻辑拉，男的讲的是有理逻辑，:).因此你还是学无理代数吧 -rabbitbug(兔八哥); 2003-1-24 (#998641@0)

是我大学里分数最差的课程之一，当时全班50%要补考。:) -jianghongca(慎独); 2003-1-24 (#998387@0)

是我大学学的最好成绩的一们:98 -youngerbrother(http:fun.to/hongwei); 2003-1-24 (#998443@0)

大虾出招！帮帮rainrain -ppgg(聪明小子); 2003-1-24 (#998487@0)

可是毕业多年,咋记得那些代号,全还给老师了 -youngerbrother(http:fun.to/hongwei); 2003-1-24 (#998647@0)

这年头，还有人念CS，看上去还是under。搞不懂。 -wooway(纯洁中文运动[加东区]); 2003-1-24 (#998406@0)

和和，不管你搞得懂不，俺读的不是cs -rainrain(雨雨若我若无我); 2003-1-24 (#998411@0)

我猜：精算专业的选修课 -ppgg(聪明小子); 2003-1-24 (#998419@0)

好象才是集合论部分的题目嘛，小菜 -wooway(纯洁中文运动[加东区]); 2003-1-24 (#998430@0)

no, 这个小菜是代数系统。 -rainrain(雨雨若我若无我); 2003-1-24 (#998488@0)
光会说。hoho -ppgg(聪明小子); 2003-1-24 (#998492@0)

有啥搞不懂的,毕业后的形势谁说的准. -fox69(fox); 2003-1-24 (#998442@0)

我还是看好CS。——毛主席说过，世界上就怕认真二字。只要自己喜欢的专业，有点钻研精神，都是可以成功的。 -becky(爱喝冷血的妖精); 2003-1-24 (#998460@0)

CS 不等于写代码！ CS的基础理论才是最精彩的 -ppgg(聪明小子); 2003-1-24 (#998486@0)

天灵灵地灵灵，妖精神仙JJMMGGDD快现形帮俺呀 -rainrain(雨雨若我若无我); 2003-1-24 (#998634@0)
不懂，你们的数学好象和我们不一样啊，sorry.：（（（ -60years(浪子归); 2003-1-25 (#998670@0)
哎，可惜我见到数字就头疼 :-( 帮你UP一下吧. -ilovetomato(怕冷的青番茄); 2003-1-25 (#998688@0)
离散数学偶最拿手, 当年考第一, 考试的时候旁边都是请我吃饭的,现在忘的差不多了. 如果有SLX, 可考虑重抄旧业. ^)* -smallrainrain(小雨雨俺是酒仙); 2003-1-25 (#998706@0)

老小，我记得你上次给俺弟妹to be做过题的？快点帮帮俺吧，尤其是第1个和第2个，新的俺没听课，旧的都忘光了:( -rainrain(雨雨若我若无我); 2003-1-25 (#998724@0)

吼吼, 你不睡偶可要睡了, 今天加班到晚上10点, 明天白天还要去,如果你不急, 明天晚上帮你研究吧. SORRY. ^)& -smallrainrain(小雨雨俺是酒仙); 2003-1-25 (#998737@0)

第一题是要说明是群。要证四点：存在一元，存在逆元，运算封闭，满足结合律 -flyingover(flyingover); 2003-1-25 (#998774@0)

我明天试试，谢谢！！！ -rainrain(雨雨若我若无我); 2003-1-25 (#998814@0)

做出了第1道，其它2个还是不会：（很光火 -rainrain(雨雨霏来霏去); 2003-1-26 (#1002344@0)

以做饭当诱饵，就会有N个哥哥排队上门做功课了。 -asdfasdf(牛牛很牛); 2003-1-26 (#1002377@0)
第2题很容易的说, 你要我做了发给你么? ^_* -smallrainrain(小雨雨俺是酒仙); 2003-1-26 (#1002404@0)

老小，第2个第一问俺经过下面同学的提示弄出来了:( 第2问怎么证明呀？~~~@_@ -rainrain(雨雨霏来霏去); 2003-1-26 (#1002780@0)

Some hints. -rumba(rumba); 2003-1-26 {805} (#1002712@0)
2.1 Show nZ satisfy the four axioms under + operation, you can use the condition that (Z,+) is a group.
2.2 If m not exist, then S can not form a group.
3.1 Z is set of integer, (Z,+) form a group, identity is 0, inverse is negative.
Z4 is mod(Z,4) = {0,1,2,3}. (Z4,+) form a group, it preserves the identity (0) and operation (+) as group (Z,+), so f is homeomorphism. However, the inverse on (Z4, +) is 4-a if the original one is a.
3.2 Equivalent class of any integer x on Z of mod 4 operation is: sx = { ...., x-8, x-4, x, x+4, x+8, ....}.
To prove p is an equivalent relation, 1) Reflexive. sx = sx. 2) Symmetric. sx1 = sx2 ==> sx2 = sx1. 3) Transitive.
All distince equivalent class: {...x-4, x, x+4, ...} {...,x-5, x-1, x+3, ...} {...,x-6,x-2,x+2,...} {...,x-7,x-3,x+1,...}

多谢乐，明天早上就due,俺现在就try。 -rainrain(雨雨霏来霏去); 2003-1-26 (#1002785@0)

离散数学包括：集合论、图论、代数体系（近世代数）等部分，我当年可是考A的哦，可是我现在不能告诉你，因为 -jqian(Q_Q); 2003-1-27 {30} (#1004860@0)
这么多年了，从来不用，忘了。:D

@Calgary

Some hints.

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