Page 1 of 2 12 LastLast
Results 1 to 15 of 16

Math Help - Proof: Section- Finite and Infinite Sets

  1. #1
    Newbie
    Joined
    Mar 2011
    Posts
    10

    Proof: Section- Finite and Infinite Sets

    Hello, this is for my Advanced Calculus Class.

    If A, B, and C are sets, then prove...

    a) A ~ A
    b) A ~ B if and only if B ~ A
    c) A ~ B and B ~ C implies A ~ C


    I need to prove each one, I tried working on each but I just end up getting lost or messing it up and it doesn't make sense. Please prove them.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Mush89 View Post
    If A, B, and C are sets, then prove...
    a) A ~ A
    b) A ~ B if and only if B ~ A
    c) A ~ B and B ~ C implies A ~ C
    This is a perfect example of your needing to include more detail when posting. It may surprise you to know that A\sim A may mean anything. I taught advanced for thirty years. I have never seen that notation.
    So tell us the meaning and definition of all relevant terms.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2011
    Posts
    10
    "~" means equivalent, or equipotent. It's used all over the Math books I've seen.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Mush89 View Post
    "~" means equivalent, or equipotent. It's used all over the Math books I've seen.
    How does your textbook define equipotent?
    Again there are several different approaches to this topic.
    If we have to guess, it may just confuse you more.
    Please be detailed when posting questions.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Aug 2010
    Posts
    885
    Thanks
    91
    Quote Originally Posted by Mush89 View Post
    Hello, this is for my Advanced Calculus Class.

    If A, B, and C are sets, then prove...

    a) A ~ A (symmetry)
    b) A ~ B if and only if B ~ A (reflexivity)
    c) A ~ B and B ~ C implies A ~ C (transitivity)


    I need to prove each one, I tried working on each but I just end up getting lost or messing it up and it doesn't make sense. Please prove them.
    I added items in parentheses to above quote.

    Shilov uses \sim to denote equivalence, and defines equivalence to be a one-to-one correspondence between sets. In that case, symmetry and reflexivity are obvious, and for transitivity:
    Corresponding to a in A there is a unique b in B and corresponding to b in B there is a unique c in C so corresponding to a in A there is a unique c in C. And then the other way around.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Hartlw View Post
    I added items in parentheses to above quote.
    a) A ~ A (symmetry)
    b) A ~ B if and only if B ~ A (reflexivity)
    c) A ~ B and B ~ C implies A ~ C (transitivity)
    Shilov uses \sim to denote equivalence, and defines equivalence to be a one-to-one correspondence between sets.
    Hartlw, you have once again jumped in without getting it right.
    You have the two terms in red above backwards.
    Moreover, you do not know that the OP have with Shilov's way of defining this relation.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Mar 2011
    Posts
    10
    This proof is for my Advanced Calculus class. It's in a section called Finite and Infinite Sets. I need to prove a, b, c if A, B, and C are sets.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Mush89 View Post
    This proof is for my Advanced Calculus class. It's in a section called Finite and Infinite Sets. I need to prove a, b, c if A, B, and C are sets.
    OK. You wrote that A\sim B in your text material means that A\text{ is equipotent to }B .
    Now tell us how the textbook defines equipotent.
    At this point we are still just guessing.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Mar 2011
    Posts
    10
    It says, "Two sets, A and B, are equivalent, or equipotent, denoted by A ~ B, if and only if a bijection exists from A onto B. And equipotent meaning they have the same cardinality.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1
    Quote Originally Posted by Mush89 View Post
    It says, "Two sets, A and B, are equivalent, or equipotent, denoted by A ~ B, if and only if a bijection exists from A onto B. And equipotent meaning they have the same cardinality.
    Thank you. Had that been included to begin with, you would have gotten real help sooner;.
    f:x\mapsto x the identity mapping is a bijection of any set to itself. Thus, A\sim A.

    If f:A\to B is a bijection then f^{-1}:B\to A is also a bijection. Thus we have symmetry.

    If f:A\to B~\&~g:B\to C are both bijections then g\circ f:A\to C is also a bijection.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Super Member
    Joined
    Aug 2010
    Posts
    885
    Thanks
    91
    Quote Originally Posted by Plato View Post
    Thank you. Had that been included to begin with, you would have gotten real help sooner;.
    f:x\mapsto x the identity mapping is a bijection of any set to itself. Thus, A\sim A.

    If f:A\to B is a bijection then f^{-1}:B\to A is also a bijection. Thus we have symmetry.

    If f:A\to B~\&~g:B\to C are both bijections then g\circ f:A\to C is also a bijection.
    This was already done in above post by Hartlw. From Wofram Math World:

    "Two sets A and B are said to be equipollent iff there is a one-to-one correspondence (i.e., a bijection) from A onto B (Moore 1982, p. 10; Rubin 1967, p. 67; Suppes 1972, p. 91)."
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Quote Originally Posted by Hartlw View Post
    This was already done in above post by Hartlw. From Wofram Math World:

    "Two sets A and B are said to be equipollent iff there is a one-to-one correspondence (i.e., a bijection) from A onto B (Moore 1982, p. 10; Rubin 1967, p. 67; Suppes 1972, p. 91)."
    Plato's point was that there are several different (equivalent) ways to define equipotency, and without knowing which way the author is referring to, we can not really help him.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Super Member
    Joined
    Aug 2010
    Posts
    885
    Thanks
    91
    Quote Originally Posted by Defunkt View Post
    Plato's point was that there are several different (equivalent) ways to define equipotency, and without knowing which way the author is referring to, we can not really help him.
    The only one I could find wrt two sets A and B is one-to-one onto. Do you know of any others?
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Yes - an equivalent definition is that there is an injection f: A \to B and an injection g: B \to A.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Super Member
    Joined
    Aug 2010
    Posts
    885
    Thanks
    91
    Quote Originally Posted by Defunkt View Post
    Yes - an equivalent definition is that there is an injection f: A \to B and an injection g: B \to A.
    From Wolfram Math World (google "injection"):

    "In other words, is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one."

    However, the conditions a), b), and c) of OP can only be satisfied if it is one-to-one-onto, which I used in my proof, ie, we are still talking about the same thing, one-to-one onto.

    Speaking of my original proof, I identified a) and b) of OP as symmetry and reflexivity, when they are conventionally identified the other way around, except by Shilov (pg 30), which I happened to have at hand and because I can't remember which is which and A \sim A looks symmetric to me- I guess it makes more sense that way to us Russians. Technically, and from a purist point of view, having defined my terms my proof was correct within the context of the definition, and, btw, also for the reverse definition, and it really didn't depend on how I happened to identify conditions a) and b) anyhow. Its a triviality which produced a very harsh response from Plato, and a warning message from the powers that be.
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Finite and infinite sets
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: August 6th 2011, 03:53 PM
  2. Real Analysis: Union of Countable Infinite Sets Proof
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: September 14th 2010, 05:38 PM
  3. Contradiction Proof- injection, Infinite Sets
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 24th 2010, 10:20 AM
  4. finite and infinite sets...denumerable
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: November 22nd 2009, 05:38 PM
  5. Proof: finite sets
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: February 26th 2009, 10:01 PM

Search Tags


/mathhelpforum @mathhelpforum