- #1
dkavlak
- 2
- 0
Hello,
So I've been running into problems with rigorously proving that a function I've defined in ZFC is a bijection (1-1 and onto).
For example, if I know that a function between two numbers "n" and "m" (defined in the standard von neumann way) is a bijection (call the function "f"), how can I use "f" to prove that a function "g" such that g={<x,y>| <x,y> is in f or <x,y> = <{n},{m}>} (equivalently, g is a function between the successor of n and the successor of m) is a bijection?
It seems to me obviously true. Since there is a bijection between n and m, surely if you add one element to n and add one element to m you can construct a bijection between the new sets (since the two sets have cardinality there will be a way to map each distinct element of one set to a distinct element of the other and vice versa).
I realize this is a very elementary question, but it is something I keep struggling to accomplish and would greatly appreciate someone helping me to understand the proof technique needed. Equally helpful would be an explanation of how to prove that any function you define is a bijection.
Thanks for your help.
So I've been running into problems with rigorously proving that a function I've defined in ZFC is a bijection (1-1 and onto).
For example, if I know that a function between two numbers "n" and "m" (defined in the standard von neumann way) is a bijection (call the function "f"), how can I use "f" to prove that a function "g" such that g={<x,y>| <x,y> is in f or <x,y> = <{n},{m}>} (equivalently, g is a function between the successor of n and the successor of m) is a bijection?
It seems to me obviously true. Since there is a bijection between n and m, surely if you add one element to n and add one element to m you can construct a bijection between the new sets (since the two sets have cardinality there will be a way to map each distinct element of one set to a distinct element of the other and vice versa).
I realize this is a very elementary question, but it is something I keep struggling to accomplish and would greatly appreciate someone helping me to understand the proof technique needed. Equally helpful would be an explanation of how to prove that any function you define is a bijection.
Thanks for your help.