The mapping is f:N →N, given by F(n)= 2n^3=1,n belongs to N.
A. one-one and onto
B. onto not one-one
C. one-one not onto
D. neither one-one nor onto
To determine whether a mapping is one-one (injective) and/or onto (surjective), we need to consider the definition and properties of these terms.
A mapping is one-one (injective) if each element in the domain maps to a unique element in the codomain. In other words, no two different elements in the domain can map to the same element in the codomain.
A mapping is onto (surjective) if every element in the codomain is mapped to by at least one element in the domain. In other words, there are no elements in the codomain that are not mapped to by any element in the domain.
Now, let's analyze the given mapping:
The mapping is F(n) = 2n^3 - 1, where n belongs to N (the set of natural numbers).
To determine if the mapping is one-one (injective):
We can use the method of contradiction. Let's assume there exist two distinct natural numbers, a and b, such that F(a) = F(b).
So, 2a^3 - 1 = 2b^3 - 1.
By canceling out the common terms, we get a^3 = b^3.
Taking the cube root on both sides, we have a = b.
This contradicts our assumption that a and b are distinct, implying that the mapping is one-one (injective).
To determine if the mapping is onto (surjective):
We need to check if every element in the codomain (N) can be obtained from the mapping.
Let's say we want to find an element in N, y, such that it is mapped to by some element in the domain. We can solve the equation 2n^3 - 1 = y for n.
Rearranging the equation, we get 2n^3 = y + 1.
Dividing both sides by 2, we have n^3 = (y + 1) / 2.
Taking the cube root of both sides, we find n = (y + 1)^(1/3) / 2.
It is worth noting that (y + 1)^(1/3) is not always a natural number. Therefore, there may be elements in the codomain that do not have a corresponding element in the domain.
Hence, the mapping is not onto (not surjective).
In conclusion, based on our analysis:
The given mapping f(n) = 2n^3 - 1, where n belongs to N, is one-one (injective) but not onto (not surjective).
Therefore, the correct option is C. one-one not onto.