Q. 19 C5.0( 2 Votes )

Give an example of a map

which is neither one-one nor onto.

Answer :

Let the function f :R R be defined by f (x) = cosx, x R.


We have,


f (x) = cosx, x R


In order to prove that f is one-one, it is sufficient to prove that f(x1)=f(x2) x1=x2 x1, x2 A .


Let x1 = 0 and x2 = 2π are two different elements in R.


Now,


f(x1) = f(0) = cos0 = 1


f(x2) = f(2π) = cos2π = 1


we observe that f(x1)=f(x2) but x1 ≠ x2.


This shows that different element in R may have same image.


Thus, f(x) is not one-one.


We know that cosx lies between -1 and 1.


So, the range of f is [-1,1] which is not equal to its co-domain.


i.e., range of f ≠ R (co-domain)


In other words, range of f is less than co-domain, i.e there are elements in co-domain which does not have any pre-image in domain.


so, f is not onto.


Hence, f is neither one-one nor onto.


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