anonymous 5 years ago what is a function? just a simple question. who can answer?

1. amistre64

a function is defined as for any one input; there is only one output that can be produced

2. amistre64

a relation exists between any given sets of data; but a function is useful in that we can use it to predict events

3. anonymous

ahaa tnx

4. anonymous

Let A and B be sets of numbers. A function f is a rule that assigns each element of A to a unique element of B.

5. anonymous

what if B has infinitely many element from A ., is that can be called as a function?

6. amistre64

If A results in many B's then a relation exists; but not a function

7. amistre64

If B results from multiple A's, then a function can be determined to best match the results

8. anonymous

*an

9. anonymous

how is it done?

10. anonymous

give all the functions. A={1,2,3} B={d,f}

11. anonymous

There are infinitely many.

12. anonymous

prove it.

13. amistre64

are you mapping A into B?

14. amistre64

f:A -> B

15. anonymous

f:A->B

16. anonymous

Let A={x: x=1,2 or 3} and B={d or f}. Then$f:A \rightarrow B$ where f can be any function.

17. anonymous

Can you show it for me all the functions?

18. anonymous

No. And here are some corrections: B={g(x): g(x)=d or f} and g: A -> B.

19. anonymous

Suppose that the set C containing all g such that g: x -> g(x) (where x is in A and g(x) is in B) is finite. Then there exists a h in C such that h: x -> h(x), where h(x) is in B. But h1= h(x)+1-1 is also in C, and so is h2=h1+1-1, and so is h3=h2+1-1, . . . Therefore, C is infinite. This is a contradiction. Therefore, C must be infinite.

20. anonymous

And therefore there are infinitely many g.

21. anonymous

No: wait. I have made an error. Two function are the same if their domains (A in this case), codomains (B), and effects are the same. Thus h=h1=h2= . . . However, I still cannot list all functions g by the same argument as in the above comment. Can you see why?