Re: set question

In article <1170105616.564786.285570@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<bikrant@xxxxxxxxx> wrote:
Hi
Can anyone please let me know if there exists bijection function from=20
=D8 to =D8. I think it does. But I am not able to find a convincing reason=20
to it.

Please stick to ASCII (plain text) characters. Your message was
garbled for many readers (myself among them).

You are asking if there is a bijection from the empty set to the empty
set.

The answer is yes.

Because everything in empty set maps to the other empty set ?


Yes; because "everything in the empty set" has Property P is always
true.

Put another way: remember that a function f from a set A to a set B is a
subset of A x B that satisfied the following two properties:

(i) For every a in A there exists b in B such that (a,b) is in f;
and
(ii) For every a in A, and b, c in B, if (a,b) and (a,c) are both in
f, then b=c.


The function is one-to-one if and only if it further satisfies that

(iii) For every a,a' in A, and b in B, if (a,b) and (a',b) are
elements of f, then a=a'.

and it is surjective if and only if in addition to (i) and (ii) it
satisfies

(iv) For every b in B there exists a in A such that (a,b) is in f.

The function is a bijection from A to B if and only if it satisfies
both (iii) and (iv) (in addition to (i) and (ii) to be a function).

Let f be the empty set. I claim that f satisfies the conditions
(i)-(iv), needed to be "a bijection from emptyset to emptyset."

Okay: for every a in emptyset, there exists b in emptyset such that
(a,b) is in f. True, by vacuity (there are no a in emptyset, so the
statement is true). Thus, (i) holds.

For every a in empty set, and b,c in empty set, if (a,b) and (a,c) are
both in f, then b=c. True, again, by vacuity: no a in emptyset. So
(ii) Holds.

(iii) again holds because A is empty, by vacuity. And (iv) holds
because B is empty, again by vacuity.

Since f = emptyset satisfies the necessary conditions, it is a
bijection from emptyset to emptyset.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: does the empty set contain the least member?
    ... If you mean that you are considering the empty set as a partially ... ordered set with the empty order relation, then one needs to ask if it ... This is true because "x notin emptyset" is true for all x, ... which is a statement that is true by vacuity for ...
    (sci.math)
  • Re: Definition of functions.
    ... (Arturo Magidin) ... to the above definition the empty set is a bijection between the empty set ... ordered triple, since the range of f is a subset of B, and the ... bijection, ...
    (sci.math)
  • Re: Definition of functions.
    ... I must admit I usually pay ... to the above definition the empty set is a bijection between the empty set ... The proper definition of a function is ... bijection, ...
    (sci.math)
  • Re: Definition of functions.
    ... I seem to recall seeing thatdefinition. ... to the abovedefinitionthe empty set is a bijection between the empty set ...
    (sci.math)
  • Re: Sets of functions
    ... >> denotes the empty set}, ... >> bijection from the empty set to itself. ... A singleton is a set with exactly one element. ... However,you suffer from not having a reader not giving you feedback on ...
    (sci.math)