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

(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 by
showing a functionf: {0,1}
2
→ {0, 1}
3
that 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 by
showing a functionf: {0,1}
2
→ {0, 1}
2
that is neither one-to-one nor onto.
4.3.4: Properties of functions on strings and power sets
For 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 an
example showing why.
(a)
f: {0, 1}
4
→ {0, 1}
3
. The output of f is obtained by taking the input string and
dropping 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 and
replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For
example, 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 and
reversing the bits. For example f(011) = 110.

Solution
One-to-one, but not onto. The output string always has the property that the
first 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