Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

B Natural Numbers and Odd Numbers

  1. Jun 11, 2017 #1
    Given any finite set of natural numbers, it seems evident that the odd numbers form a subset of the natural numbers. But what happens "at infinity"? I mean, if we account for all infinitely many natural numbers, there would be also infinitely many odd numbers. In such case, is it still true that the odd ones form a subset of the natural ones?

    <Title edited along with this note about the title. fresh_42>
     
    Last edited by a moderator: Jun 11, 2017
  2. jcsd
  3. Jun 11, 2017 #2

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What's the definition of subset?
     
  4. Jun 11, 2017 #3
    Given two sets ##A## and ##B##, ##A## is said to be a subset of ##B## if ##A \cap B = A## and ##A \cap B \neq B##.
     
  5. Jun 11, 2017 #4

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I wasn't expecting that!

    I might prefer the following:

    ##A## is a subset of ##B## if ##x \in A \ \Rightarrow \ x \in B##
     
  6. Jun 11, 2017 #5
    :smile:
    So, since by definition any positive odd number is also a natural number, we conclude that the odd positive numbers form a subset of the natural numbers even when we have infinitely many numbers?
     
  7. Jun 11, 2017 #6

    fresh_42

    Staff: Mentor

  8. Jun 11, 2017 #7

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes.
     
  9. Jun 11, 2017 #8
  10. Jun 11, 2017 #9

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Usually a subset can be the full set: ##A \subseteq A##. In this case the second condition is not satisfied.

    If the second condition is satisfied, it is a proper subset.

    Edit: Better symbol.
     
    Last edited: Jun 11, 2017
  11. Jun 11, 2017 #10
    This is a bit nonsense. But definitions are definitions.
     
  12. Jun 11, 2017 #11

    fresh_42

    Staff: Mentor

    The only nonsense is that he should have written ##A \subseteq A ## instead of ##A \subset A##, but the rest and the main idea is correct. ##A \subseteq A## is a subset. ##A \subsetneq B## is a proper subset. The additional condition ##A \cap B \neq B## is unusual as long as one doesn't define a proper subset. To exclude equality makes the entire topic only unnecessarily complicated, IMO.
     
  13. Jun 11, 2017 #12

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Many mathematical statements would have to use "a subset of A or A itself" everywhere if you exclude the full set as subset.

    It is like the convention that 1 is not a prime number. Otherwise you have "for every prime apart from 1" everywhere.
     
  14. Jun 11, 2017 #13

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I've always liked this convention as I find something very satisfying about:

    A = B iff A is a subset of B and B is a subset of A.

    Having to say "subset or equal to" would spoil that.
     
  15. Jun 11, 2017 #14

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    I don't know if just a convention; if 1 were prime, every number would be composite as n=n(1).
     
  16. Jun 11, 2017 #15

    fresh_42

    Staff: Mentor

    If a unit was prime the entire concept would be meaningless. We have this (IMO senseless) discussion only because they learn at school "if only divisible by ##1## and itself". If they learnt it correctly, this wouldn't be necessary.
     
  17. Jun 11, 2017 #16

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Yes, but I ( think I ) get mfb's point that , by strict definition, 1 is a(n) ( integer) prime, since it is divisible only by itself...and by 1.
     
  18. Jun 11, 2017 #17

    fresh_42

    Staff: Mentor

    But ##1## is neither irreducible (in ##\mathbb{Z}##) nor does ##1 \mid ab## imply ##1 \mid a## or ##1 \mid b##. It only happens both to be true as for every unit. Why did never ever ask anyone, why ##-1## isn't prime? It simply contradicts the idea behind it.
     
  19. Jun 11, 2017 #18

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Still, I guess is the rype of thing that needs to be clarified just once, after which one can move on.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Natural Numbers and Odd Numbers
  1. Natural Numbers (Replies: 7)

  2. Natural Numbers (Replies: 3)

  3. Natural Numbers (Replies: 11)

Loading...