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

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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).

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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)}

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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 b • a 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 !)

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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.

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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=n^{2} 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 n^{m} 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.

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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

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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").

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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.

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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)