anonymous
  • anonymous
plearse teach me injective, surjective and byjective
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
@zzr0ck3r
zzr0ck3r
  • zzr0ck3r
Do you know what a function is?
anonymous
  • anonymous
yes

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anonymous
  • anonymous
you are given variables and replaced by numbers for short
zzr0ck3r
  • 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?
anonymous
  • anonymous
no. please explain
zzr0ck3r
  • 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
anonymous
  • anonymous
ok. so the first is injective?
zzr0ck3r
  • zzr0ck3r
correct
zzr0ck3r
  • zzr0ck3r
the second one is not because 1 and 2 both get sent to a
anonymous
  • anonymous
ok
anonymous
  • anonymous
in any function, every x maps to at most one y in the range
zzr0ck3r
  • zzr0ck3r
A function is onto (surjective) if everything in the codomain gets used up example |dw:1441760739646:dw| non example |dw:1441760788586:dw|
anonymous
  • anonymous
read this http://www.math.ucla.edu/~tao/java/MultipleChoice/functions.txt
anonymous
  • anonymous
ok
anonymous
  • 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.
zzr0ck3r
  • 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
zzr0ck3r
  • zzr0ck3r
Ok @GIL.ojei Is the following function surjective, injective both, or neither? |dw:1441761073116:dw| ?
anonymous
  • anonymous
injective
zzr0ck3r
  • zzr0ck3r
why?
zzr0ck3r
  • 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
anonymous
  • 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
zzr0ck3r
  • zzr0ck3r
@GIL.ojei why?
anonymous
  • 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
anonymous
  • anonymous
because each maps differently and are all exhausted in the left
zzr0ck3r
  • 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.
zzr0ck3r
  • zzr0ck3r
Ok @GIL.ojei is it surjective
anonymous
  • anonymous
gil no two x values map to the same y value. agreed?
anonymous
  • anonymous
yes
zzr0ck3r
  • zzr0ck3r
OK @GIL.ojei is it surjective?
anonymous
  • anonymous
no
zzr0ck3r
  • zzr0ck3r
why ot?
zzr0ck3r
  • zzr0ck3r
remember that surjective just means that the entire codomain (the circle on the right) gets used up
anonymous
  • anonymous
because no single element of X mspd to two elements of Y
zzr0ck3r
  • zzr0ck3r
read my last comment
zzr0ck3r
  • zzr0ck3r
what you just said is true of all functions
zzr0ck3r
  • 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
anonymous
  • anonymous
oh ok
zzr0ck3r
  • zzr0ck3r
ok so when it is both surjective and injective we call it a bijection.
zzr0ck3r
  • 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.
anonymous
  • anonymous
ok
zzr0ck3r
  • zzr0ck3r
Now think about why these things are saying what we talked about above.
zzr0ck3r
  • 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
anonymous
  • anonymous
ok buy i do not understand this
anonymous
  • 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.
zzr0ck3r
  • zzr0ck3r
buy ?
anonymous
  • anonymous
but
zzr0ck3r
  • 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 \(f(a) = f(b) \implies a=b\)
anonymous
  • anonymous
ok
zzr0ck3r
  • zzr0ck3r
does that make sense?
zzr0ck3r
  • 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.
anonymous
  • anonymous
yes thanks
anonymous
  • anonymous
when will you be online again sir?
zzr0ck3r
  • zzr0ck3r
most likely

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