Surjective and/or injective? Please show all work completely, thank you. Let ome
ID: 3034660 • Letter: S
Question
Surjective and/or injective?
Please show all work completely, thank you.
Let omega be the word, omega = supercahfragilisticexpialidocious, and let X be the set of letters, X = {a, c, d, e, f, g, i, l, o, p, r, s, t, u, x} and let y be the set of numbers: Y = {y Z | 1 lessthanorequalto y lessthanorequalto 34}. Define the function, f: X rightarrow Y, as f(x) = the number of occurrences of the letter, "x", in the word, omega. Is f and injective function? Is f a surjective function? Give a brief explanation for each answer.Explanation / Answer
The given function f : XY is neither injective nor surjective.
a) injective : a function f(x) is said to be injective or one-one if the images of distinct elements of set X under f are distinct in set Y. But in our case, f(x) is many-one. For example : albhabets (d, f, g, o, r, t, x) have same occurrence 1, means non-distinct images in set Y.
b) surjective : a function f(x) is said to be surjective or onto if every element of set Y is the image of some elements of set X under f. Means for every y 'belongs to' set Y, there exists an element x in set X such that y = f(x). But in our case, f(x) is not onto. For example : y = 5, 6, 8, .... 34 in set Y are not images of any element x of set X.
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