Question

plearse teach me injective, surjective and byjective

Answer 1

@zzr0ck3r

Answer 2

Do you know what a function is?

Answer 3

yes

Answer 4

you are given variables and replaced by numbers for short

Answer 5

A function is one-to-one (injective) if every x in the domain maps to at most one y in the codomain

example



here is a non example



Does this make sense?

Answer 6

no. please explain

Answer 7

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

Answer 8

ok. so the first is injective?

Answer 9

correct

Answer 10

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

Answer 11

ok

Answer 12

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

Answer 13

A function is onto (surjective) if everything in the codomain gets used up

example



non example

Answer 14

read this http://www.math.ucla.edu/~tao/java/MultipleChoice/functions.txt

Answer 15

ok

Answer 16

" 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.

Answer 17

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

Answer 18

Ok @GIL.ojei

Is the following function surjective, injective both, or neither?



?

Answer 19

injective

Answer 20

why?

Answer 21

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

Answer 22

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

Answer 23

@GIL.ojei why?

Answer 24

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

Answer 25

because each maps differently and are all exhausted in the left

Answer 26

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.

Answer 27

Ok @GIL.ojei is it surjective

Answer 28

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

Answer 29

yes