1. Let \( A = \{ x, y, z \} \) (a) If possible, give an example of a subset of \(\mathcal{P}(A)\) with cardinality 4. (b) If possible, give an example of an element of \(\mathcal{P}(A)\) with cardinality 4. (c) Does there exist a function \( f : A \rightarrow \mathcal{P}(A) \) which is 1-1? If such a function exists, give an example; if no such function exists, explain why.

Given set is A=x, y, z P(A) is the power set of A. This is the collection…

Answered: 1. Let A= {x, y, z} (a) If possible,…

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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Determine whether these statements are true or false. Enter T or F. (a) Cardinality of {Ø} is 0. (b) {0}{0} (c) Ø C {0} (d) 0 € Ø
Let f {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} → {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be defined by f(x) = 2x if x 6. Select all the elements of f({5,6}). 01 3 2 4 Select all the elements of f-¹({5,6}). 3 04 Select all the elements of ƒ−¹(ƒ({5,6})). 3 04 2 Select all the elements of ƒ(ƒ¯¹({5,6})). 01 3 04 5 5 06 6 07 07 07 07 U O 8 8 8 8 9 9 9 ✔10 10 10 10
Does there exist a 1-1 and onto function that maps the integers to the rationales? Choose the best response below. Yes, but the function is not easily written in a "normal" way. Yes, and thus you can easily write it explicitly. No, because you can't write it explictly. No because the two set don't have the same cardinality.
For every element r in A, there is an element y in B such that (x, y) is in F. Can you explain how does this differ from a relation? Please answer this in a couple sentences NO NEED TO SHOW PROOF! AGAIN NO PROOF PLEASE!!!
Suppose f: A --> B is a function. Define the function F: P(B) --> P(A) (from the powerset of B to the one of A) by setting F(S) = f-1(S) for all S subset of B. Prove or disprove: (1) F is 1-1 if and only if f is onto. (2) F is onto if and only if f is 1-1. [Finite, not infinite.]
How to prove the following: if A ⊂ image f then f(f -1(A)) = A.
How do I prove this? Please include detailed explanation!
13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.
2. Prove that [0,1] and U=1 An , where A, = [2n, 2n+1] have the same cardinality. Please note that in this assignment, it is not enough to simply give a function and state that it is injective/surjective/bijective. You have to provide a PROOF that the function you're providing has these properties.

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