# B if a function f is one to one then the target of f

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(b)If a function f is one-to-one, then the target of f is at least as large as its domain.Show that the converse of the previous statement is not necessarily true byshowing a functionf: {0,1}2{0, 1}3that is not one-to-one.
(c)If a function f is a bijection, then the domain of f is the same size as its target.Show that the converse of the previous statement is not necessarily true byshowing a functionf: {0,1}2{0, 1}2that is neither one-to-one nor onto.
4.3.4: Properties of functions on strings and power setsFor each of the functions below, indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give anexample showing why.(a)f: {0, 1}4{0, 1}3. The output of f is obtained by taking the input string anddropping the first bit. For example f(1011) = 011.
(b)f: {0, 1}3{0, 1}3. The output of f is obtained by taking the input string andreplacing the first bit by 1, regardless of whether the first bit is a 0 or 1. Forexample, f(001) = 101 andf(110) = 110.
(c)f: {0, 1}3{0, 1}3. The output of f is obtained by taking the input string andreversing the bits. For example f(011) = 110.
SolutionOne-to-one, but not onto. The output string always has the property that thefirst bit is the same as the last bit, so there is no x, for example, such that f(x) =1000.(e)Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8}. f: P(A){0, 1, 2, 3, 4, 5, 6,7, 8}. ForX ⊆ A, f(X) = |X|. Recall that for a finite set A, P(A) denotes the power

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Term
Spring
Professor
Meenakshisundaram
Tags
Finite set, Bijection