## anonymous one year ago plearse teach me injective, surjective and byjective

1. anonymous

@zzr0ck3r

2. zzr0ck3r

Do you know what a function is?

3. anonymous

yes

4. anonymous

you are given variables and replaced by numbers for short

5. zzr0ck3r

A function is one-to-one (injective) if every x in the domain maps to at most one y in the codomain example |dw:1441760363692:dw| here is a non example |dw:1441760400010:dw| Does this make sense?

6. anonymous

7. zzr0ck3r

notice that in the first picture each element in the domain (the one on the left) gets sent to only one element on the right (this is the codomain) Notice this is NOT the case in the non example

8. anonymous

ok. so the first is injective?

9. zzr0ck3r

correct

10. zzr0ck3r

the second one is not because 1 and 2 both get sent to a

11. anonymous

ok

12. anonymous

in any function, every x maps to at most one y in the range

13. zzr0ck3r

A function is onto (surjective) if everything in the codomain gets used up example |dw:1441760739646:dw| non example |dw:1441760788586:dw|

14. anonymous
15. anonymous

ok

16. anonymous

" For every x in X there is at most one y in Y such that f(x) = y." Comment. Every function f has this property (they each map one element to one element, i.e. they are not "one-to-two"). However, this is not what one-to-one means.

17. zzr0ck3r

A function must have 2 properties 1) it is defined everywhere i.e. everything in the domain gets used up notice how surjectivity is sort of like the opposite of this 2) it is well defined i.e. each x can map to at most one y notice how injectivity is the opposite of this

18. zzr0ck3r

Ok @GIL.ojei Is the following function surjective, injective both, or neither? |dw:1441761073116:dw| ?

19. anonymous

injective

20. zzr0ck3r

why?

21. zzr0ck3r

@jayzdd I think it is much better to start with a intuitive notion. this notation is not going to help imo. that comes next

22. anonymous

a better definition for injectivity distinct (different) inputs map to distinct outputs formally: if a ≠ b, then f(a) ≠ f(b) this is equivalent to if f(a) = f(b), then a = b

23. zzr0ck3r

@GIL.ojei why?

24. anonymous

yes you're diagrams are good to explain the intuition. i was taking issue with the phrase 'every x has at most one y ' for your one to one thats true about all functions

25. anonymous

because each maps differently and are all exhausted in the left

26. zzr0ck3r

because each maps differently, is why it is injective The fact that each gets used up is actually a property of it being a function.

27. zzr0ck3r

Ok @GIL.ojei is it surjective

28. anonymous

gil no two x values map to the same y value. agreed?

29. anonymous

yes

30. zzr0ck3r

OK @GIL.ojei is it surjective?

31. anonymous

no

32. zzr0ck3r

why ot?

33. zzr0ck3r

remember that surjective just means that the entire codomain (the circle on the right) gets used up

34. anonymous

because no single element of X mspd to two elements of Y

35. zzr0ck3r

36. zzr0ck3r

what you just said is true of all functions

37. zzr0ck3r

It is injective because no two x elements get sent to one y value It is surjective because every element in the codomain gets used up

38. anonymous

oh ok

39. zzr0ck3r

ok so when it is both surjective and injective we call it a bijection.

40. zzr0ck3r

Here is what is important, and here is the notation that we use A function $$f:D\rightarrow C$$ is a relation with the following two properties. 1) $$a=b\implies f(a)=f(b)$$ 2) $$\forall x\in D$$ it is true that $$f(x)\in C$$ A function is surjective if $$f(a)=f(b)\implies a=b$$ where $$a,b\in D$$. A function is surjective if $$\forall y\in C \ \exists \ x\in D$$ such that $$f(x) = y$$. A function that is both surjective and injective is called a bijection.

41. anonymous

ok

42. zzr0ck3r

Now think about why these things are saying what we talked about above.

43. zzr0ck3r

@jayzdd pointed out, in my very first post I should have said injective implies no two domain elements map to a single codomain element

44. anonymous

ok buy i do not understand this

45. anonymous

A function $$f:D\rightarrow C$$ is a relation with the following two properties. 1) $$a=b\implies f(a)=f(b)$$ 2) $$\forall x\in D$$ it is true that $$f(x)\in C$$ A function is surjective if $$f(a)=f(b)\implies a=b$$ where $$a,b\in D$$. A function is surjective if $$\forall y\in C \ \exists \ x\in D$$ such that $$f(x) = y$$. A function that is both surjective and injective is called a bijection.

46. zzr0ck3r

47. anonymous

but

48. zzr0ck3r

Ok well lets talk about the surjective surjective says that no two different domain elements can map to the same y value, so if two things map to the same y value they better be the same thing in other words $$f(a) = f(b) \implies a=b$$

49. anonymous

ok

50. zzr0ck3r

does that make sense?

51. zzr0ck3r

The reason we are going over this part is because you are going to be asked to show a function is surjective and injective and this is how you do it.

52. anonymous

yes thanks

53. anonymous

when will you be online again sir?

54. zzr0ck3r

most likely