Exercise 11. Let B be a finite set, let f be a function on B, and let A = f[B]. Prove that A is finite and Ā≤B. (Hint: Use Theorem 13.38, Theorem 13.40, and Corollary 13.31.)
Exercise 11. Let B be a finite set, let f be a function on B, and let A = f[B]. Prove that A is finite and Ā≤B. (Hint: Use Theorem 13.38, Theorem 13.40, and Corollary 13.31.)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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![Exercise 11. Let B be a finite set, let f be a function on B, and let A = f[B]. Prove that A is finite
and Ā≤ Ē. (Hint: Use Theorem 13.38, Theorem 13.40, and Corollary 13.31.)
13.38 Theorem. Let B be a finite set and let f be a function on B. Then f has a right inverse. In other
words, there is a function g: A → B, where A = f[B], such that for each x € A, we have f(g(x)) = x.
13.40 Theorem. A right inverse for a function is an injection.
13.31 Corollary. Let B be a finite set and let A be a set. Then (a) and (b) below are equivalent.
(a) A is equinumerous to a subset of B.
(b) A is finite and Ā≤Ē.
12.22 Definition. Let f be a function and let A be a subset of the domain of f. Then the image of A
under f (denoted f[A]) is the set of all values that f takes on at points of A; in other words,
ƒ[A] = {f(x) : x € A}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a262c0f-e9c3-46ae-bb60-dfa33b1b3cf6%2Fe11db161-8ec9-4557-a825-a9eeead71112%2Fo9equi8_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 11. Let B be a finite set, let f be a function on B, and let A = f[B]. Prove that A is finite
and Ā≤ Ē. (Hint: Use Theorem 13.38, Theorem 13.40, and Corollary 13.31.)
13.38 Theorem. Let B be a finite set and let f be a function on B. Then f has a right inverse. In other
words, there is a function g: A → B, where A = f[B], such that for each x € A, we have f(g(x)) = x.
13.40 Theorem. A right inverse for a function is an injection.
13.31 Corollary. Let B be a finite set and let A be a set. Then (a) and (b) below are equivalent.
(a) A is equinumerous to a subset of B.
(b) A is finite and Ā≤Ē.
12.22 Definition. Let f be a function and let A be a subset of the domain of f. Then the image of A
under f (denoted f[A]) is the set of all values that f takes on at points of A; in other words,
ƒ[A] = {f(x) : x € A}.
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