Spivak chapter 3 question 6

  • #1
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1
Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it:

For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance.

I get this isn't really calculus but its in the book so I thought the responders here would be most useful.

I appreciate all responses.
 

Answers and Replies

  • #2
101
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I have moved forward while I wait for a response and have come to problem 9 with characteristic functions and it asks me to express A intersects B, X is either in A or B, and R - A ( all as subsets of C) in terms of [itex]C_A[/itex] and [itex]C_B[/itex].

I don't know what it is asking me to do.
 
  • #3
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What is [itex]f[/itex]? What's its domain? Is it continuous?

If it's continuous and its domain includes zero, those might be useful properties. If there's also a point [itex]x>0[/itex] such that [itex]f(x)=f(0)[/itex], that might also be a useful property.
 
  • #4
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For the second question, it's asking for algebraic ways of writing the statements given in terms of [itex]C_A,C_B[/itex]. For example, the statement [tex]A\neq\emptyset[/tex] is equivalent to the expression [itex]C_A\neq 0[/itex].
 
  • #5
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Thanks. I didn't think of it that way.
 
  • #6
pasmith
Homework Helper
1,777
445
Hello, I have been backtracking recently on Spivak problems to ensure I am fluent and I'm doing this one and I just cannot solve it:

For which numbers C is there a number X such that f(cx) = f(x). Hint: there are a lot more than you might think at first glance.
Given c, all you need to do is choose x such that cx = x.
 
  • #7
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The domain is x cannot equal -1. How does that help me? You could say that it is any number if x=0. Is that acceptable?

For the second problem, I still don't see what you mean. How do I algebraically manipulate them?
 
  • #8
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I have just realised this is in the wrong space but i don't know how to move it. Im sorry and feel free to move it.
 
  • #9
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It turned out that C can be anything so i was right there. The second one is a bit tricky though. I will reread the text first.
 
  • #10
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I see that [itex]C_A[/itex][itex]\bullet[/itex][itex]C_B[/itex] or [itex]C_A[/itex][itex]+[/itex][itex]C_B[/itex] could both be the answer for [itex]C_A\cap C_B[/itex] as both intersect with x. Is this right?How do i tell which it is?
 
  • #11
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Economics nerd, you're saying that you can say A is not empty by writing [itex]C_A[/itex] is not 0 as, if its zero, x is not in A so A is empty.

I see what you mean but could you please explain to me how these characteristic functions work. I get that C_A(x) means that there is an x for every A. However, I am having relating that to [itex]C_A \cup C_B[/itex]?
 
  • #12
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Any response to my two questions?
 
  • #13
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Also, in this context, what is the difference between adding, multiplying and subtracting in terms of the domain.
 
  • #15
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Is it normal to have issues with such problems early on, even if it is spivak?
 
  • #16
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Anybody got any hints?
 

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