How to show two sets are equal

What we need to do:

This is equivalent to proving:

• an element belongs to A if and only if it belongs to B, or else
• (i) A is a subset of B, and (ii) B is a subset of A

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  What we say:

  Let x be an element of A...
  

  ...and so x belongs to B. Now, let x be an element of B...

  

  ...and so x belongs to A.

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  What we want to do:

  Use the information about the sets to see what you know about the element you pick. (What does belonging to A tell you about x?) Use the description of B to see what you need to do to prove that x belongs to B. (What must x satisfy in order for us to know that it belongs to B?)

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  Example:

  Question

  

  Solution

  

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  Analysis WebNotes by John Lindsay Orr.
  Comments to the author: jorr@math.unl.edu

  All contents copyright (C) 1995 John L. Orr
  University of Nebraska--Lincoln
  All rights reserved

  Last modified: June 13, 1995