91Ó°ÊÓ

(a) Draw an arrow diagram that represents a function that is an injection but is not a surjection. (b) Draw an arrow diagram that represents a function that is an injection and is a surjection. (c) Draw an arrow diagram that represents a function that is not an injection and is not a surjection. (d) Draw an arrow diagram that represents a function that is not an injection but is a surjection. (e) Draw an arrow diagram that represents a function that is not a bijection.

For each of the following functions, determine if the function is an injection and determine if the function is a surjection. Justify all conclusions. (a) \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f(x)=3 x+1,\) for all \(x \in \mathbb{Z}\). (b) \(F: \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(F(x)=3 x+1,\) for all \(x \in \mathbb{Q}\) (c) \(g: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(g(x)=x^{3},\) for all \(x \in \mathbb{R}\) (d) \(G: \mathbb{Q} \rightarrow \mathbb{Q}\) defined by \(G(x)=x^{3},\) for all \(x \in \mathbb{Q}\). (e) \(k: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(k(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}\) (f) \(K: \mathbb{R}^{*} \rightarrow \mathbb{R}\) defined by \(K(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}^{*}\). Note: \(\mathbb{R}^{*}=\\{x \in \mathbb{R} \mid x \geq 0\\}\). (g) \(K_{1}: \mathbb{R}^{*} \rightarrow T\) defined by \(K_{1}(x)=e^{-x^{2}},\) for all \(x \in \mathbb{R}^{*},\) where \(T=\) \(\\{y \in \mathbb{R} \mid 0

Let \(s: \mathbb{N} \rightarrow \mathbb{N},\) where for each \(n \in \mathbb{N}, s(n)\) is the sum of the distinct natural number divisors of \(n\). This is the sum of the divisors function that was introduced in Preview Activity 2 from Section 6.1. Is \(s\) an injection? Is \(s\) a surjection? Justify your conclusions.

Let \(d: \mathbb{N} \rightarrow \mathbb{N},\) where \(d(n)\) is the number of natural number divisors of \(n\). This is the number of divisors function introduced in Exercise (6) from Section \(6.1 .\) Is the function \(d\) an injection? Is the function \(d\) a surjection? Justify your conclusions.

Let \(f: A \rightarrow B\) and \(g: B \rightarrow A\). Let \(I_{A}\) and \(I_{B}\) be the identity functions on the sets \(A\) and \(B\), respectively. Prove each of the following: (a) If \(g \circ f=I_{A},\) then \(f\) is an injection. (b) If \(f \circ g=I_{B},\) then \(f\) is a surjection. (c) If \(g \circ f=I_{A}\) and \(f \circ g=I_{B},\) then \(f\) and \(g\) are bijections and \(g=f^{-1}\)

The number of divisors function. Let \(d\) be the function that associates with each natural number the number of its natural number divisors. That is, \(d: \mathbb{N} \rightarrow \mathbb{N}\) where \(d(n)\) is the number of natural number divisors of \(n\). For example, \(d(6)=4\) since \(1,2,3,\) and 6 are the natural number divisors of 6 (a) Calculate \(d(k)\) for each natural number \(k\) from 1 through 12 . (b) Does there exist a natural number \(n\) such that \(d(n)=1 ?\) What is the set of preimages of the natural number \(1 ?\) (c) Does there exist a natural number \(n\) such that \(d(n)=2 ?\) If so, determine the set of all preimages of the natural number \(2 .\) (d) Is the following statement true or false? Justify your conclusion. For all \(m, n \in \mathbb{N},\) if \(m \neq n,\) then \(d(m) \neq d(n) .\) (e) Calculate \(d\left(2^{k}\right)\) for \(k=0\) and for each natural number \(k\) from 1 through 6 (f) Based on your work in Exercise (6e), make a conjecture for a formula for \(d\left(2^{n}\right)\) where \(n\) is a nonnegative integer. Then explain why your conjecture is correct. (g) Is the following statement is true or false? For each \(n \in \mathbb{N},\) there exists a natural number \(m\) such that \(d(m)=n\)

For each of the following, give an example of functions \(f: A \rightarrow B\) and \(g: B \rightarrow C\) that satisfy the stated conditions, or explain why no such example exists. "(a) The function \(f\) is a surjection, but the function \(g \circ f\) is not a surjection. (b) The function \(f\) is an injection, but the function \(g \circ f\) is not an injection. (c) The function \(g\) is a surjection, but the function \(g \circ f\) is not a surjection. (d) The function \(g\) is an injection, but the function \(g \circ f\) is not an injection. (e) The function \(f\) is not a surjection, but the function \(g \circ f\) is a surjection. (f) The function \(f\) is not an injection, but the function \(g \circ f\) is an injection. (g) The function \(g\) is not a surjection, but the function \(g \circ f\) is a surjection. (h) The function \(g\) is not an injection, but the function \(g \circ f\) is an injection.

In Exercise (6), we introduced the number of divisors function \(d\). For this function, \(d: \mathbb{N} \rightarrow \mathbb{N},\) where \(d(n)\) is the number of natural number divisors of \(n\) A function that is related to this function is the so-called set of divisors function. This can be defined as a function \(S\) that associates with each natural number the set of its distinct natural number factors. For example. \(S(6)=\\{1,2,3,6\\}\) and \(S(10)=\\{1,2,5,10\\}\) (a) Discuss the function \(S\) by carefully stating its domain, codomain, and its rule for determining outputs. (b) Determine \(S(n)\) for at least five different values of \(n\). - (c) Determine \(S(n)\) for at least three different prime number values of \(n\). (d) Does there exist a natural number \(n\) such that card \((S(n))=1 ? \mathrm{Ex}-\) plain. [Recall that card \((S(n))\) is the number of elements in the set \(S(n) .]\) (e) Does there exist a natural number \(n\) such that card \((S(n))=2 ? \mathrm{Ex}=\) plain. (f) Write the output for the function \(d\) in terms of the output for the function \(S\). That is, write \(d(n)\) in terms of \(S(n)\). (g) Is the following statement true or false? Justify your conclusion. For all natural numbers \(m\) and \(n,\) if \(m \neq n,\) then \(S(m) \neq S(n)\). (h) Is the following statement true or false? Justify your conclusion. For all sets \(T\) that are subsets of \(\mathbb{N}\), there exists a natural number \(n\) such that \(S(n)=T\)

Creating Functions with Finite Domains. Let \(A=\\{a, b, c, d\\}, B=\) \(\\{a, b, c\\},\) and \(C=\\{s, t, u, v\\} .\) In each of the following exercises, draw an arrow diagram to represent your function when it is appropriate. (a) Create a function \(f: A \rightarrow C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (b) Create a function \(f: A \rightarrow C\) whose range is the set \(\\{u, v\\}\) or explain why it is not possible to construct such a function. (c) Create a function \(f: B \rightarrow C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (d) Create a function \(f: A \rightarrow C\) whose range is the set \(\\{u\\}\) or explain why it is not possible to construct such a function. (e) If possible, create a function \(f: A \rightarrow C\) that satisfies the following condition: For all \(x, y \in A,\) if \(x \neq y,\) then \(f(x) \neq f(y)\) If it is not possible to create such a function, explain why. (f) If possible, create a function \(f: A \rightarrow\\{s, l, u\\}\) that satisfies the following condition: For all \(x, y \in A,\) if \(x \neq y,\) then \(f(x) \neq f(y) .\) If it is not possible to create such a function, explain why.

Is the following proposition true or false? Justify your conclusion with a proof or a counterexample. If \(f: S \rightarrow T\) is an injection and \(A \subseteq S,\) then \(f^{-1}(f(A))=A\).