Let M = {5, 6, 7, 8} and N = {3, 4, 9, 10}. Which one of the following functions is neither one-one nor onto?a)f = {(5, 3), (5, 4), (6, 4), (8, 9)}b)f = {(5, 3), (6, 4), (7, 9), (8, 10)}c)f = {(5, 4), (5, 9), (6, 3), (7, 10), (8, 10)}d)f = {(6, 4), (7, 3), (7, 9), (8, 10)}Correct answer is option 'A'. Can you explain this answer?

Question

Answer

The function f = {(5, 3), (5, 4), (6, 4), (8, 9)} is neither one-one nor onto.
The function is not one – one 8 does not have an image in the codomain N and we know that a function can only be one – one if every element in the set M has an image in the codomain N.
A function can be onto only if each element in the co-domain has a pre-image in the domain X. In the function f = {(5, 3), (5, 4), (6, 4), (8, 9)}, 10 in the co-domain N does not have a pre-image in the domain X.
f = {(5, 3), (6, 4), (7, 9), (8, 10)} is both one-one and onto.
f = {(5, 4), (5, 9), (6, 3), (7, 10), (8, 10)} and f = {(6, 4), (7, 3),(7, 9), (8, 10)} are many – one onto.

Answer

Explanation:
To determine whether a function is one-one or onto, we need to understand the definitions of these terms:

- One-one function: Also known as an injective function, a one-one function is a function in which each element of the domain maps to a distinct element in the co-domain. In other words, no two elements in the domain can map to the same element in the co-domain.

- Onto function: Also known as a surjective function, an onto function is a function in which every element in the co-domain has a corresponding element in the domain that maps to it. In other words, the range of the function is equal to the co-domain.

Analysis of the given options:
Let's analyze each option to determine if it is one-one or onto:

a) f = {(5, 3), (5, 4), (6, 4), (8, 9)}
- This function is not one-one because the element 5 in the domain maps to both 3 and 4 in the co-domain.
- This function is not onto because the element 7 in the co-domain does not have a corresponding element in the domain that maps to it.

b) f = {(5, 3), (6, 4), (7, 9), (8, 10)}
- This function is one-one because each element in the domain maps to a distinct element in the co-domain.
- This function is onto because every element in the co-domain has a corresponding element in the domain that maps to it.

c) f = {(5, 4), (5, 9), (6, 3), (7, 10), (8, 10)}
- This function is not one-one because the element 5 in the domain maps to both 4 and 9 in the co-domain.
- This function is not onto because the element 3 in the co-domain does not have a corresponding element in the domain that maps to it.

d) f = {(6, 4), (7, 3), (7, 9), (8, 10)}
- This function is not one-one because the element 7 in the domain maps to both 3 and 9 in the co-domain.
- This function is not onto because the element 5 in the co-domain does not have a corresponding element in the domain that maps to it.

Conclusion:
Based on the analysis, option a) f = {(5, 3), (5, 4), (6, 4), (8, 9)} is neither one-one nor onto.

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