(c) Prove that if A and B are finite sets, then A B is finite and A B| = |B|Al. (Hint: Use induction on [A|.) (d) A professor has 20 students in his class, and he has to assign a grade of either A, B, C, D, or F to each student. In how many ways can the grades be assigned?
(c) Prove that if A and B are finite sets, then A B is finite and A B| = |B|Al. (Hint: Use induction on [A|.) (d) A professor has 20 students in his class, and he has to assign a grade of either A, B, C, D, or F to each student. In how many ways can the grades be assigned?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
just parts c and d
Transcribed Image Text:**Exercise 23:**
**For any sets \( A \) and \( B \), the set of all functions from \( A \) to \( B \) is denoted \( B^A \).**
**(a)** Prove that if \( A \sim B \) and \( C \sim D \) then \( C^A \sim D^B \).
**(b)** Prove that if \( A \), \( B \), and \( C \) are sets and \( A \cap B = \varnothing \), then \( C^{A \cup B} \sim C^A \times C^B \).
**(c)** Prove that if \( A \) and \( B \) are finite sets, then \( B^A \) is finite and \(|B^A| = |B|^{|A|}\). (Hint: Use induction on \(|A|\).)
**(d)** A professor has 20 students in his class, and he has to assign a grade of either A, B, C, D, or F to each student. In how many ways can the grades be assigned?
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