Set of a which is the set of all subsets of a

set of A which is the set of all subsets of A.
(f)
Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8}. f: P(A) → P(A). For X ⊆
A, f(X) = X. Recall that for a finite set A, P(A) denotes the power set of A which is
the set of all subsets of A.
Solution
One-to-one and onto.
(g)
Let A be defined to be the set {1, 2, 3, 4, 5, 6, 7, 8} and let B = {1}. f: P(A)
P(A). ForX ⊆ A, f(X) = X - B. Recall that for a finite set A, P(A) denotes the power
set of A which is the set of all subsets of A.
→
4.4.1: Arrow diagrams for inverses of functions
Each of the arrow diagrams below define a function f. For each arrow diagram,
indicate whether f
-1
is well-defined. If f
-1
is not well-defined, indicate why. If f
-1
is
well-defined, give an arrow diagram showing f
-1
.
(a)
(b)

Solution
f
-1
is well-defined.
(c)
4.4.2: Finding inverses of functions
For each of the following functions, indicate whether the function has a well-
defined inverse. If the inverse is well-defined, give the input/output relationship
of f
-1
.
(a)
f: Z → Z . f(x) = x + 3
(b)
f: Z → Z . f(x) = 2x + 3

(c)
f: R → R . f(x) = 2x + 3
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
set of A which is the set of all subsets of A.