3. Let o: {1,2, 3, ...} → {1,2, 3, ...} denote the Euler phi-function. If ø injective? Is it surjective? Justify your answers.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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МCBS
МАT 255
Note: For Problems 3 and 4, you may use the following two facts about the Euler Phi-function:
a) If integers m and n are coprime, then o(mn) = ¢(m)o(n).
b) If p is a prime number and a is a positive integer, then o(p") = pª-1(p – 1)
1. In a class of 16 students, there are 6 male students and 10 female students. In how many ways can
you select a group of 5 students from this class such that 3 of the 5 students are males and the other
two are females?
2. Use the inclusion-exclusion principle to find the number of solutions to the equation
x + y + z = 19,
where x, y, and z are positive integers. [Hint: You know how to find the number of non-negative
solutions to the equation above.]
3. Let ø: {1, 2, 3, ...} → {1, 2, 3, ...} denote the Euler phi-function. If ø injective? Is it surjective? Justify
your answers.
4. Define a relation - on the set N of natural numbers as follows: For natural numbers n and m, n ~ m if
and only if ø(n) = ¢(m), where o denotes the Euler phi-function. Is ~ defined as such an equivalence
relation on N? If so, find the equivalence class containing 5.
5. Let S be a set containing 8 elements. What is the number of subsets of S containing at least two
elements? Justify your answer.
Transcribed Image Text:МCBS МАT 255 Note: For Problems 3 and 4, you may use the following two facts about the Euler Phi-function: a) If integers m and n are coprime, then o(mn) = ¢(m)o(n). b) If p is a prime number and a is a positive integer, then o(p") = pª-1(p – 1) 1. In a class of 16 students, there are 6 male students and 10 female students. In how many ways can you select a group of 5 students from this class such that 3 of the 5 students are males and the other two are females? 2. Use the inclusion-exclusion principle to find the number of solutions to the equation x + y + z = 19, where x, y, and z are positive integers. [Hint: You know how to find the number of non-negative solutions to the equation above.] 3. Let ø: {1, 2, 3, ...} → {1, 2, 3, ...} denote the Euler phi-function. If ø injective? Is it surjective? Justify your answers. 4. Define a relation - on the set N of natural numbers as follows: For natural numbers n and m, n ~ m if and only if ø(n) = ¢(m), where o denotes the Euler phi-function. Is ~ defined as such an equivalence relation on N? If so, find the equivalence class containing 5. 5. Let S be a set containing 8 elements. What is the number of subsets of S containing at least two elements? Justify your answer.
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