Answers to problems

: 136.330 Intro to Algebra: Answers to problems
   By Anonymous on Tuesday, September 17, 2002 - 09:05 pm: Edit Post

Hi! i'm wondering how to do 2.8., 2.21,2.22


   By Sasho on Tuesday, September 17, 2002 - 11:11 pm: Edit Post

Please write down the statements of the problems (for the sake of the people who want to see both the statement and a solution or a hint, and because I do not drag the textbook with me and so I do not know what, say, 2.8 is).


   By Anonymous on Thursday, September 19, 2002 - 12:57 pm: Edit Post

2.8 Describe the inverse of the mapping Beta in Example 2.1

Example 2.1
Let S={x,y,z} T={1,2,3} and U={a,b,c}
Define theta:S-T by theta(x)=2 theta(y)=1 theta(z)=3
beta(1)=b, beta(2)=c beta(3)=a

2.21 give an example of sets s,t, and and mappings theta: s-t and beta: t-u such beta*theta is onto, but theta is not onto.[compare theorem 2.1(b)}

2.22 give an example of sets s,t, and and mappings theta: s-t and beta: t-u such beta*theta is onto, but beta is not one to one.[compare theorem 2.1(b)}


   By Sasho on Thursday, September 19, 2002 - 01:49 pm: Edit Post

2.8.
The inverse b-1 of b: T ® U is the mapping U ®T that works as follows: b-1(a)=3, b-1(b)=1 and b-1(c)=2 (just follow in the opposite direction the arrows in the textbook describing b).

2.21.

Take S={1}, T={a,b}, U={x} and a(1)=a, b(a)=x=b(b). Then ba is certainly onto, but a is not onto.

2.22.

The example in 2.21 will do the work for 2.22 as well ! (check !)


   By Paul on Tuesday, October 01, 2002 - 12:05 pm: Edit Post

There are a couple of examples in the book that I am having problems with. If you could please help me out it would be greatly appreciated.

page 14 #1.21a),b)
Assume that S and T are finite sets containing m and n elements, respectively.
a) How many mappings are there form S to T?
B) How many one-to-one mappings are there from S to T?

#1.29
Prove that a mapping alpha: S-T is one-to-one iff alpha(A intersection B) = alpha(A) intersection alpha(B) for every pair of subsets A and B of S.


   By Sasho on Tuesday, October 01, 2002 - 01:58 pm: Edit Post

14. (a) the first element of S could go to any of the n elements of T (n choices); given where the first element of S is mapped, the second could still be mapped to any of the n elements of T; we have n.n=n2 many choices for the images of the first two elements of S. Keep going untill we exhaust all of the elements of S, to get the final ansewer of nm mappings S®T.

(b) requires a similar argument. The final answer is either 0 if m>n, or n(n-1)(n-2)...(n-m+1) if n is at least as large as m.

1.29. I wil show a part of one of the two implications: if a:S®T is one-to-one, then a(A intersect B) = a(A) intersect a(B).

So, suppose a: is 1-1. Take an element x in a(A intersect B). Then x=a(y) for some y in A intersect B. But since y is in A intersect B, we have that y is both in A and in B. So, x=a(y) is in a(A) (since y is in A), and x=a(y) is in a(B) (since y is in B). So, since we got that x is in both a(A) and a(B), it must be in a(A) intersect a(B). This shows that a(A intersect B) is a subset of a(A) intersect a(B). You then prove that a(A) intersect a(B) is a subset of a(A intersect B) by taking an element in the first set and showing it must be in the second. That would do this implication. [Note that I have not used the assumption that a is 1-1; it is needed in the part that is omitted (that a(A) intersect a(B) is a subset of a(A intersect B))].

Then you do the converse (if a(A intersect B) = a(A) intersect a(B) then a is 1-1).

Here is something to make your life easier: I will not give you that question for the test - it needs too much of manual work. But, a small portion of it might be in.


   By Paul on Monday, October 28, 2002 - 10:35 pm: Edit Post

I am having a few difficulties with equivalence classes. If you could please answer the following questions, I should be able to figure the rest out.

q. 9.8 b)
Define a relation ~ on the set N of natural numbers by a~b iff a = (b)(10^k) for some k in Z.
Give a complete set of equivalence class representatives.

q. 9.10
Give a complete set of equivalence class representatives for the equivalence relation in Example 9.2. (Let L denote the set of all lines in a plane with rectangular coordinate system. For l1, l2 in L, let l1~l2 mean that l1 and l2 have equal slopes or that both slopes are undefined. This is an equivalence relation on L. The set of lines equivalent to a line l consists of l and all lines in L that are parallel to l.)

Thanks


   By Sasho on Tuesday, October 29, 2002 - 09:57 am: Edit Post

9.8(b) Observe that a~b if a=b00000 (a bunch of 0-s) or if b=a00000 (a bunch of 0-s). So, the equivalence classes are {1,10,100,1000,...}, {2,20,200,...}, ....{9,90,900,...}, {11,110,1100,...}, {12,120,...},.... (representatives of these classes are numbers that do NOT end with 0).

9.10. The last sentence in the statement almost gives the answer to the question: one equivalence class is the set of lines of a fixed slope r; as r ranges over the set of all real numbers we get almost all equivalence classes; the only one that is missing and should also be accounted for is the equivalence class of all vertical lines (of slope "infinity").


   By Paul on Tuesday, October 29, 2002 - 04:14 pm: Edit Post

Hello again,
You gave us a proof to do in class a while ago, order of An is 1/2(n!). I took a look in the book but I don't understand what they are doing with theta. If you could please clear that up for me, thanks.


   By Sasho on Tuesday, October 29, 2002 - 04:42 pm: Edit Post

I dont have the textbook with me; so you need to write down the part that is unclear to you.

And by the way - please start a new topic; this one is becoming too long. (use "create new conversation" button at the bottom of the page listing the topics so far)


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