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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)
asked in Mathematical Logic by Veteran (11.8k points) 13 43 139 | 117 views
I think all are representing one-to-one function.

1 Answer

+4 votes

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..

answered by Veteran (89.1k points) 15 58 295
Why $S_3$ is false?
@habib thnx for rply
can u plzz tell me what is wrong in S3 and S4. ??
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..
@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.


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