Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Concept explainers

Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question

just parts c and d

**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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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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