1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove that {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} ⊂ {x∈ℤ : 6|x}

  1. Jan 31, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} ⊂ {x∈ℤ : 6|x}.
    It means that the intersection of the two sets is subset of the set containing integers that are divisible by 6.

    3. The attempt at a solution
    Basically I need to show that if an integer is divisible by 2 and 9, then it's also divisible by 6.
    Proof:
    let. x=2*9*n=18*n=(6*3*n)∈{x∈ℤ : 6|x} (n∈ℤ)

    Q.E.D.

    However, I don't think it's that simple.
     
    Last edited: Jan 31, 2016
  2. jcsd
  3. Jan 31, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    You think wrong: it is correct. :oldsmile:
     
  4. Jan 31, 2016 #3
    Ok, thanks :oldsmile:
     
  5. Jan 31, 2016 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Actually, it's correct but needs some justification. Suppose I said x is divisible by 2 and 6. You would be wrong to say that x=2*6*n and conclude that's it's divisible by 12. You need to say something about prime numbers.
     
  6. Feb 1, 2016 #5
    Let's summarize the proof.

    I have already shown that if x is even, then the statement is true.

    If x is a prime, additive inverse of a prime or ±1, then {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} = the empty set. And the empty set is a subset of every set. In addition, all primes are odd.

    Therefore {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} ⊂ {x∈ℤ : 6|x} holds for all x∈ℤ.
     
    Last edited: Feb 1, 2016
  7. Feb 1, 2016 #6

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    I'm a little lost here. The statement "If x is a prime, additive inverse of a prime or ±1, then {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} = the empty set" is meaningless, or overuses the symbol "x".

    What Dick meant was (I think) that you should explain more in detail why you can write x=2*9*n in your original proof. He gave as an example that this would not work for 2 and 6.
     
  8. Feb 1, 2016 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Right, that's what I meant. x being prime isn't an issue, it's something about 2 and 3 being prime.
     
  9. Feb 1, 2016 #8
    Is it about prime factors? 2*3*3*n? How should I modify the original proof?
     
  10. Feb 1, 2016 #9

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Not by much, as both Dick and myself said it is correct.
    What you haven't explicitly explained is the following:
    Take x ∈{y∈ℤ : 2|y} ∩ {y∈ℤ : 9|y}.
    That says that x=2*a and x=9*b, where a and b are integers.
    How do you conclude from this that x=2*9*n (n an integer)? We saw that for the pair 2,6 that doesn't work.
     
  11. Feb 1, 2016 #10
    Isn't that obvious? I still dont get it.

    x=2*a and x=9*b
    So a=9 and b=2, because x is an integer.
     
  12. Feb 1, 2016 #11

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    It is rather obvious, but the point was that you didn't state why.

    Let's assume I don't get it: I don't see why x=2*a and x=9*b implies x=2*9*n (a, b, n integers). How would you explain it to me?

    This is not correct.
     
  13. Feb 1, 2016 #12
    If x is divisible by 2 and 9, then x must be product of at least 2 and 9.
     
  14. Feb 1, 2016 #13

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Why?
    Just start from x=2*a=3*b (a,b integers).
    EDIT: sorry, that whas a typo, as above it should have ben x=2*a=3*3*b
     
    Last edited: Feb 1, 2016
  15. Feb 1, 2016 #14
    x=2*a=3*b
    a=3b/2
     
  16. Feb 1, 2016 #15

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    Let's try this: we have as before x=2*a=9*b.

    Do you think that b can be an odd integer? And why?
     
  17. Feb 1, 2016 #16

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The basic theorem you want to apply here is called "unique prime factorization". Samy A suggests an alternative trick way to do it. But I think you should work with the theorem. Unless you don't know it, in which case the trick is good.
     
    Last edited: Feb 1, 2016
  18. Feb 2, 2016 #17

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    I agree, and I assume he knows the theorem. But it looks as if he sees the theorem as so obvious that he doesn't actually states it.
     
  19. Feb 2, 2016 #18
    let x=2*3*3*n=6*3n?
     
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: Prove that {x∈ℤ : 2|x} ∩ {x∈ℤ : 9|x} ⊂ {x∈ℤ : 6|x}
  1. How to prove -(-x)=x (Replies: 21)

Loading...