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State with reason whether 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)}` |
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Answer» (i) `f: {1, 2, 3, 4} → {10}` defined as: `f = {(1, 10), (2, 10), (3, 10), (4, 10)}` From the given definition of f, we can see that f is a many one function as:` f(1) = f(2) = f(3) = f(4) = 10` `:.` f is not one-one. Hence, function f does not have an inverse. (ii) `g: {5, 6, 7, 8} → {1, 2, 3, 4}` defined as: `g = {(5, 4), (6, 3), (7, 4), (8, 2)}` From the given definition of `g`, it is seen that g is a many one function as: `g(5) = g(7) = 4`. `:. g` is not one-one, Hence, function g does not have an inverse. (iii) `h: {2, 3, 4, 5} → {7, 9, 11, 13}` defined as: `h = {(2, 7), (3, 9), (4, 11), (5, 13)}` It is seen that all distinct elements of the set `{2, 3, 4, 5}` have distinct images under ` h`. `:.` Function h is one-one. Also, h is onto since for every element `y` of the set `{7, 9, 11, 13}`, there exists an element `x` in the set `{2, 3, 4, 5}` such that `h(x) = y`. Thus, `h` is a one-one and onto function. Hence, `h` has an inverse. |
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