Function

Question

Which of the following statement/s representing One-to-One Function.

S1; ∀a∀b(f (a) = f (b) → a = b)

S2: ∀a∀b( a ≠ b→f (a) ≠ f (b) )

S3:  ∀a∀b(a = b → f (a) = f (b)),

S4:  ∀a∀b(f (a) ≠ f (b) → a ≠ b)

Comments

I think all are representing one-to-one function.

Answer 1

From the definition of one one function :

 The function f is injective(synonym of one-one) if and only if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b.  Equivalently, if a ≠ b, then f(a) ≠ f(b).

So if we represent them in logic form , we get S1 is immediate logical translation of an statement..And since implication represented by S1 is true , so is its contrapositive which is represented by S2 ..

So S1 and S2 are valid logical propositions regarding one one functions..

Comments

Why $S_3$ is false?

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@habib thnx for rply
can u plzz tell me what is wrong in S3 and S4. ??

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This is just the mathematical definition of one one function which is one way implication..

And we know converse of one way implication is not necessarily true..

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@Habib. every statement is half correct . It has to be:

(a=b) ==> f(a)=f(b)    ^  (a$\neq$b) ==> f(a)$\neq$f(b)

Else, I can have one-many functions as well.