Results 1 to 2 of 2

Math Help - Can an empty set be involved in a 1-1 correspondence?

  1. #1
    Junior Member
    Joined
    Dec 2007
    Posts
    25

    Can an empty set be involved in a 1-1 correspondence?

    Say A is empty.

    Could I do A --> B (1-1) or B --> A (1-1)?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Chizum View Post
    Say A is empty.

    Could I do A --> B (1-1) or B --> A (1-1)?

    Thanks.
    Every mapping f:\varnothing\mapsto X is injective. To see this merely note that for the condition that there exists x_1,x_2\in\text{Dom}(f) such that f(x_1)=f(x_2) we would have to say x_1,x_2\in\varnothing. See a problem?

    For f:X\mapsto\varnothing it actually reverts to the same thing. For suppose that X\ne\varnothing then for any x\in X we would have that f(x) does not exist? Why? Thus, if f:X\mapsto\varnothing and f is well-defined then by necessity we must have that X=\varnothing.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. One-to-one correspondence between set of homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 13th 2012, 04:30 PM
  2. Correspondence Theorem
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 5th 2010, 07:37 PM
  3. one to one correspondence
    Posted in the Discrete Math Forum
    Replies: 17
    Last Post: October 21st 2009, 03:56 AM
  4. one to one correspondence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 12th 2009, 12:48 PM

Search Tags


/mathhelpforum @mathhelpforum