NIELIT 2016 MAR Scientist C - Section C: 16

Question

If $f:\{a,b\}^{\ast}\rightarrow \{a,b\}^{\ast }$ be given by $f(n)=ax$ for every value of $n\in \{a,b\}$, then $f$ is

• A. one to one not onto
• B. one to one and onto
• C. not one to one and not onto
• D. not one to one and onto

Comments

Note that domain is {a,b}* which is {a,b,aa,ab,ba,bb,..} and f is defined only for a and b as n belongs to {a,b} , so f is not even a function. Are you talking about partial function of f? otherwise n should belong to {a,b}*.

Answer 1

Considering x belongs to {a,b}*. Domain is {a,b,aa,ab,ba,bb,...} and range is also {a,b,aa,ab,ba,bb,...} . Suppose x1 and x2 are different elements in domain which map to the same element in range then f(x1)=f(x2)

implies a*x1=a*x2

implies x1=x2. This means x1 and x2 are same i.e. no two elements in domain will map to the same element in range. So f is one-one.

Now, a function is said to be onto if every element in range set has a pre image in domain set(no element in range set should be left out)

Here,f is not onto because for element b in range, there is no pre-image in domain.

So answer should be A

Comments

That is definition of ONTO. we cant do anything about it. Example , say domain={1,2} Range={1,2,3} then the function f:{(1,1),(2,2)} is a one one function because each element of domain is mapped to only single element of range. But it is not onto because 3 is in Range which has no pre image.

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Okk ...I know the definitions of onto but my doubt

1)f(n) = ax ,

what is x???

2) f: {a,b}* -> {a,b}*

Every to every... It's means all possible value present...

If I am wrong...plz correct

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Yes i saw that, i am assuming it is f(x)=ax otherwise f(n)=ax would mean a constant function. Every element in domain will map to same element ax and moreover ax doesnt belong to range of f. This question needs correction