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Question -

State with reason whether the following functions have inverse:
(i) f: {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4,10)}

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3),(7, 4), (8, 2)}

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3,9), (4, 11), (5, 13)}



Answer -

(i) Given f: {1, 2, 3,4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

We have:

f (1) = f (2) = f (3) = f (4) = 10

 f is notone-one.

 f is not abijection.

So, f does not have an inverse.

(ii) Given g: {5, 6,7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

from the question it is clear that g (5) = g (7) = 4

f is notone-one.

f is not abijection.

So, f does not have an inverse.

(iii) Given h: {2, 3,4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Here, different elements of the domain have different images in the co-domain.

h is one-one.

Also, each element in the co-domain has a pre-image in the domain.

 h is onto.

 h is abijection.

Therefore h inverseexists.

 h has an inverseand it is given by

h-1 = {(7, 2), (9, 3), (11, 4), (13, 5)}

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