\"Let B be a proper subset of a set A, and let f be a bijection from A to B. Pro
ID: 2942448 • Letter: #
Question
"Let B be a proper subset of a set A, and let f be a bijection from A to B. Prove that A is an infinite set. (Hint: Use Exercise 4.42)"Exercise 4.42
"Let f be a bijection from [m] to [n]. Prove that m = n. (Use induction.)"
Explanation / Answer
In order for a map f: A --> B to be a bijection, we know it must be both injective and surjective. This means that every element B must come from a unique element in A. Since B is a subset of A, by exercise 4.42 we know that A = B. This, however, contradicts the definition of a proper subset, which says in order for a subset to be a proper subset it must exclude at least one element from the group. Therefore, A cannot = B The only way for a bijection to exist is then if both are infinite sets
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