Results 1 to 15 of 15

Math Help - More sets and functions

  1. #1
    Member
    Joined
    Aug 2007
    Posts
    239

    More sets and functions

    (1) Let  f: X \to Y be a function and  A_{1}, A_{2} \in \mathcal{P}(X) . (i) Prove that  A_{1} \subseteq A_{2} \Rightarrow \overrightarrow{f}(A_{1}) \subseteq \overrightarrow{f}(A_{2}) . Prove that the converse is not universally true. Give a simple condition on  f which is equivalent to the converse. (ii) Prove that  \overrightarrow{f}(A_{1} \cap A_{2}) \subseteq \overrightarrow{f}(A_{1}) \cap \overrightarrow{f}(A_{2}) . Prove that equality is not universally true. (iii) Prove that  \overrightarrow{f}(A_{1} \cup A_{2}) = \overrightarrow{f}(A_{1}) \cup \overrightarrow{f}(A_{2}) .

    (i) So  x \in A_1 \Rightarrow x \in A_2 . Then  \overrightarrow{f}(A_{1}) = \{f(x) | x \in A_1 \} \Rightarrow \overrightarrow{f}(A_{2}) = \{f(x) | x \in A_2 \} . Thus  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) . The converse is  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) \Rightarrow A_1 \subseteq A_2 . Suppose that  A_{1} = \{c \} and  A_{2} = \{d \} . Let  f(a) = a . Then  \overrightarrow{f}(A_1) = \overrightarrow{f}(A_2) = \emptyset but  A_1 \not \subseteq A_2 . Is this correct? What would be the condition on  f which is equivalent to its converse?

    (ii)  \overrightarrow{f}(A_{1} \cap A_{2}) = \{ f(x) | x \in A_{1} \cap A_2 \} which is a subset of  \overrightarrow{f}(A_{1}) \cap \overrightarrow{f}(A_{2}) . What would be a counterexample to show inequality?

    (iii) This is basically the same as part (ii) except with an  \text{or} ?

    Thanks
    Last edited by shilz222; August 21st 2007 at 12:49 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    (i) Prove that  A_{1} \subseteq A_{2} \Rightarrow \overrightarrow{f}(A_{1}) = \overrightarrow{f}(A_{2}) .
    That is not true!

    This is true.
    (i) Prove that  A_{1} \subseteq A_{2} \Rightarrow \overrightarrow{f}(A_{1}) \subseteq \overrightarrow{f}(A_{2}) .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2007
    Posts
    239
    I meant to say that  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) .
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    I meant to say that  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) .
    Here are some notes.
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Aug 2007
    Posts
    239
    For 1(i) the converse  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) is not universally true.

    Suppose that  A_{1} = \{a, b \} and  A_2 = \{c,d \} . Define a function  f(A) = \emptyset . Then  \overrightarrow{f}(A_1) = \overrightarrow{f}(A_2) = \emptyset but  A_{1} \not \subseteq A_{2} .

    A condition on  f which is equivalent to its converse is that  f(A) \neq \emptyset .

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    Define a function  f(A) = \emptyset .
    That is not allowed.
     f(A) = \emptyset makes no sense!
    Functions are sets of ordered pairs, a subset of some A \times B,\;A \not= \emptyset  \wedge B \not= \emptyset .
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Aug 2007
    Posts
    239
    Ok if  A_ 1 = \{ a,b,c,d \} \ \text{and} \ A_2 = \{c,d,e,f,g,l,m \} and define a function  f(c) = 6 . Then  \overrightarrow{f}(A_1) \subseteq \overrightarrow{f}(A_2) = 6 but  A_1 \not \subseteq A_2 .
    Last edited by shilz222; August 23rd 2007 at 01:46 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Suppose that A = \{ 1,2,3,4\} ,\;B = \{ h,j,k,m\} then define a function f:A \mapsto B by f = \left\{ {\left( {1,j} \right),\left( {2,m} \right),\left( {3,k} \right),\left( {4,j} \right)} \right\}.
    Let A_1  = \left\{ {1,3} \right\}\;\& \;A_2  = \left\{ {2,3,4} \right\} note that \overrightarrow f \left( {A_1 } \right) \subseteq \overrightarrow f \left( {A_2 } \right)\quad but\quad A_1  \not\subseteq A_2
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Aug 2007
    Posts
    239
    Was my example correct?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    Was my example correct?
    Do you see how different in detail my example is from the one you tried to give? To be honest, I would not accept it. It lacks detail that shows that you really understand how functions operate. But I am not grading you. So I cannot say if it correct or not; just I do not find it so.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Aug 2007
    Posts
    239
    How do you become better at these types of problems and math in general? Do you just need to read a lot of books an do a lot of problems and ask interesting questions? I mean is math meant to be learned slowly?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    How do you become better at these types of problems and math in general?
    I have always advised students to simply copy out proofs from well-written textbooks, a book at the correct level. Set theory & logic is the basis of all proofs.

    Quote Originally Posted by shilz222 View Post
    I mean is math meant to be learned slowly?
    Well it certainly is not meant to be approached in a shotgun way that your assortment of problems indicates you are trying.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,888
    Thanks
    326
    Awards
    1
    Quote Originally Posted by shilz222 View Post
    How do you become better at these types of problems and math in general? Do you just need to read a lot of books an do a lot of problems and ask interesting questions? I mean is math meant to be learned slowly?
    My own take on this is that Math is learned in basically the same manner as practically any other field. You need to work problems, ask questions, and work more problems. Some time for the information to "gel" helps, as well as seeing how the material applies to either real-life problems or how it is applied to other fields.

    Just keep at it and be patient. You'll get it.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by shilz222 View Post
    I mean is math meant to be learned slowly?
    Maybe the problem is that this is very abstract to thee. The first book on math having serious proofs I ever read was on number theory which is much less abstract. And perhaps this will make you understand what proofs are about. All set theory is is doing it much much more formally.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Member
    Joined
    Aug 2007
    Posts
    239
    For 1(i) the condition on  f is that it has to be injective?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sets and functions
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 2nd 2010, 09:51 PM
  2. functions between sets
    Posted in the Discrete Math Forum
    Replies: 14
    Last Post: January 26th 2009, 08:28 PM
  3. a little bit of functions on sets
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: October 15th 2008, 06:06 PM
  4. Sets/Functions
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: October 6th 2007, 12:39 PM
  5. Sets and functions
    Posted in the Discrete Math Forum
    Replies: 12
    Last Post: August 21st 2007, 04:24 PM

Search Tags


/mathhelpforum @mathhelpforum