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Honestly I'm just having such a hard time understanding how to get this into a rigorous
fashion. I'm talking about proofs of surjectivity, injectivity & bijectivity.
The standard courses go from sets to functions to real numbers in analysis &
it's taken me quite a while to get rigorous in my proofs regarding sets. First they were
hand wavey, then they became good but extremely long requiring many many steps
to show certain things but now with the power of logic & truth table equivalences I
have reduced set proofs to two lines with full healthy understanding. I just want you
to know where I'm coming from because with functions I'm trying to get the same
pattern.
Let me give my definitions:
A mapping f | X → Y defined by f | x ↦ f(x) is the specification of a triplet (X,Y,f).
As far as I understand it the triple is an ordered set as opposed to an unordered set
{X,Y}. What I'm just curious about is while {X,Y} is an unordered set & (X,Y) is an
ordered set my book doesn't explicitly say it but is (X,Y) just another notation for the
cartesian product? They mention both things in the book one after the other but don't
make it explicit that (X,Y) is just another notation for the cartesian product but I think it's
correct. Well, assuming this is the case, what's with the notation (X,Y,f)? It's just notation
to indicate a rule is defined on the set right? (pages 9-10 of this book if you
don't know what I'm talking about).
Well a mapping f | X → Y defined by f | x ↦ f(x) is the specification of a triplet (X,Y,f).
A mapping is surjective if f(X) = Y. A mapping is injective if [f(x₁) = f(x₂)] ⇒ (x₁ = x)₂ or
equivalently [f(x₁) ≠ f(x₂)] ⇒ (x₁ ≠ x)₂.
A mapping f | X → Y is bijective if (f(X) = Y) ⋀ [ [f(x₁) ≠ f(x₂)] ⇒ (x₁ ≠ x₂) ].
Is there a more rigorous definition that hints at a proper method to prove things?
Until I found the definitions of sets involving logical implications and such I found it
very difficult to prove things regarding sets & had to rely on some intuition but now
they just roll off in a chain of logical implications which is really satisfying, I don't
see how to get that out of functions.
For instance, while I know f | R → R defining the mapping x ↦ x² is
neither injective nor surjective I know how to make it so,
f | R⁺ → R⁺ defining x ↦ x² is bijective. But if I follow the method
[f(x₁) = f(x₂)] ⇒ (x₁ = x₂) by rewriting it [f(x) = f(y)] ⇒ (x = y) like so many books
do when the mapping is f | R → R I get:
[f(x) = f(y)]
x² = y²
x = y
I mean they do it in a function like this f(x) = 3x - 7
f(x) = f(y)
3x - 7 = 3y - 7
3x = 3y
x = y
This is in a discrete math book on googlebooks as the only example in a chapter &
I mean had I not known about the methods of understanding this like I illustrated above
I'd be stumped here, I just want to get this rigorous. All the schaums manuals skip
proofs & give no helpful examples, some advanced math books I've checked on google
give no help, I really really need help here. I found one book at the weekend
describing a function with an x⁵ + 5x⁴ ... something crazy & it took a whole page to
justify the argument - something I can't find described anywhere else & I can't
find the book in my firefox history as it will take forever & I've already tried.
Any help?
fashion. I'm talking about proofs of surjectivity, injectivity & bijectivity.
The standard courses go from sets to functions to real numbers in analysis &
it's taken me quite a while to get rigorous in my proofs regarding sets. First they were
hand wavey, then they became good but extremely long requiring many many steps
to show certain things but now with the power of logic & truth table equivalences I
have reduced set proofs to two lines with full healthy understanding. I just want you
to know where I'm coming from because with functions I'm trying to get the same
pattern.
Let me give my definitions:
A mapping f | X → Y defined by f | x ↦ f(x) is the specification of a triplet (X,Y,f).
As far as I understand it the triple is an ordered set as opposed to an unordered set
{X,Y}. What I'm just curious about is while {X,Y} is an unordered set & (X,Y) is an
ordered set my book doesn't explicitly say it but is (X,Y) just another notation for the
cartesian product? They mention both things in the book one after the other but don't
make it explicit that (X,Y) is just another notation for the cartesian product but I think it's
correct. Well, assuming this is the case, what's with the notation (X,Y,f)? It's just notation
to indicate a rule is defined on the set right? (pages 9-10 of this book if you
don't know what I'm talking about).
Well a mapping f | X → Y defined by f | x ↦ f(x) is the specification of a triplet (X,Y,f).
A mapping is surjective if f(X) = Y. A mapping is injective if [f(x₁) = f(x₂)] ⇒ (x₁ = x)₂ or
equivalently [f(x₁) ≠ f(x₂)] ⇒ (x₁ ≠ x)₂.
A mapping f | X → Y is bijective if (f(X) = Y) ⋀ [ [f(x₁) ≠ f(x₂)] ⇒ (x₁ ≠ x₂) ].
Is there a more rigorous definition that hints at a proper method to prove things?
Until I found the definitions of sets involving logical implications and such I found it
very difficult to prove things regarding sets & had to rely on some intuition but now
they just roll off in a chain of logical implications which is really satisfying, I don't
see how to get that out of functions.
For instance, while I know f | R → R defining the mapping x ↦ x² is
neither injective nor surjective I know how to make it so,
f | R⁺ → R⁺ defining x ↦ x² is bijective. But if I follow the method
[f(x₁) = f(x₂)] ⇒ (x₁ = x₂) by rewriting it [f(x) = f(y)] ⇒ (x = y) like so many books
do when the mapping is f | R → R I get:
[f(x) = f(y)]
x² = y²
x = y
I mean they do it in a function like this f(x) = 3x - 7
f(x) = f(y)
3x - 7 = 3y - 7
3x = 3y
x = y
This is in a discrete math book on googlebooks as the only example in a chapter &
I mean had I not known about the methods of understanding this like I illustrated above
I'd be stumped here, I just want to get this rigorous. All the schaums manuals skip
proofs & give no helpful examples, some advanced math books I've checked on google
give no help, I really really need help here. I found one book at the weekend
describing a function with an x⁵ + 5x⁴ ... something crazy & it took a whole page to
justify the argument - something I can't find described anywhere else & I can't
find the book in my firefox history as it will take forever & I've already tried.
Any help?