Find a function \(f: \R \to \R\) such that each element in the codomain has exactly one preimage.
(b)
Find a function \(f: \R \to \R\) such that each element in the codomain has at least two preimages.
(c)
Find a function \(f: \R \to \R\) such that each element has exactly two preimages.
(d)
Find a function \(f: \R \to \R\) such that there is an element in the codomain that has exactly three preimages and another element in the codomain that as exactly two preminages.
(e)
Find a function \(f: \R \to \R\) such that there is an element in the codomain that has infinitely many preimages.
2.
For each of the following functions, determine if the function is an injection, a surjection, a bijection, or none of these. Remember to be careful about the domain and range in each case. Justify all of your conclusions.
(a)
\(F:\mathbb{R} \to \mathbb{R}\) defined by \(F( x ) = 5x + 3\text{,}\) for all \(x \in \mathbb{R}\)
(b)
\(G:\Z \to \Z\) defined by \(G( x ) = 5x + 3\text{,}\) for all \(x \in \Z\)
(c)
\(f: ( \R \setminus \left\{ 4 \right\} ) \to \R\) defined by \(f ( x ) = \dfrac{3x}{x - 4}\text{,}\) for all \(x \in \left( \R \setminus\{ 4 \} \right)\)
(d)
\(g: ( \R \setminus \left\{ 4 \right\} ) \to ( \R \setminus \left\{ 3 \right\} )\) defined by \(g ( x ) = \dfrac{3x}{x - 4}\text{,}\) for all \(x \in \left( \R \setminus \{ 4 \} \right)\)
(e)
\(h: \R \to \R_{\geq 0}\) defined by \(h(x) = x^2\) for every \(x \in \R\text{,}\) where \(\R_{\geq 0} = \{x \in \R \mid x \geq 0\}\)
(f)
\(k: \R_{\geq 0} \to \R_{\geq 0}\) defined by \(k(x) = x^2\) for every \(x \in \R_{\geq 0}\)
3.
Let \(A = \{1,2,3,4,5,6,7,8,9,10\}\) and let \(B = \{a,b,c,d,e,g\}\text{,}\) and define \(f:A \to B\) as given in Table 2.13.
Table2.13.A function from \(A\) to \(B\)
\(x\)
1
2
3
4
5
6
7
8
9
10
\(f(x)\)
\(c\)
\(d\)
\(a\)
\(g\)
\(a\)
\(c\)
\(e\)
\(d\)
\(c\)
\(a\)
(a)
If \(f\) an injection? Is \(f\) a surjection? Explain.
(b)
Find a largest subset \(C\) of \(A\) (largest in the number of elements of \(C\)) such that \(f|_C\) is an injection.
(c)
Find a subset \(D\) of \(B\) such that \(f\) is a surjection.
(d)
Find subsets \(X\) of \(A\) and \(Y\) of \(B\) such that \(f|_X : X \to Y\) is a bijection.
4.
Let \(A\) and \(B\) be sets, both of which have at least two distinct members.
(a)
Illustrate a subset \(X \subset A \times B\) that is the Cartesian product of a subset of \(A\) with a subset of \(B\text{.}\)
(b)
Show that there is a subset \(W \subset A \times B\) that is not the Cartesian product of a subset of \(A\) with a subset of \(B\text{.}\) [Thus, not every subset of a Cartesian product is the Cartesian product of a pair of subsets.]
5.
The cardinality of a finite set is defined to be the number of elements of that set. We denote the cardinality of a set \(A\) as \(|A|\text{.}\) Let \(A\) and \(B\) be sets with \(|A| = n\) and \(|B| = m\) for some positive integers \(m\) and \(n\text{.}\) Prove that there is a bijection \(f: A \to B\) if and only if \(n = m\text{.}\)
6.
Let \(X\) and \(Y\) be sets and let \(f: X \to Y\) be a function.
(a)
Let \(A\) be a subset of \(X\text{.}\) Show that \(A \subseteq f^{-1}(f(A))\text{.}\) Make an example to show that in general, \(A \neq f^{-1}(f(A))\text{.}\)
To show that the sets are not equal, consider sets \(X\) and \(Y\) with two elements.
(b)
Let \(B\) be a subset of \(Y\text{.}\) Show that \(f(f^{-1}(B)) \subseteq B\text{.}\) Make an example to show that in general, \(f(f^{-1}(B)) \neq B\text{.}\)
To show that the sets are not equal, consider sets \(X\) and \(Y\) with two elements.
(c)
Prove that \(f\) is a surjection if and only if \(f(f^{-1}(B)) = B\) for every subset \(B\) of \(Y\text{.}\)
(d)
Prove that \(f\) is an injection if and only if \(f^{-1}(f(A)) = A\) for every subset \(A\) of \(X\text{.}\)
7.
Let \(f : X \to Y\) be a function and let \(\{A_{\alpha}\}\) be a collection of subsets of \(X\) for \(\alpha\) in some indexing set \(I\text{,}\) and \(\{B_{\beta}\}\) be a collection of subsets of \(Y\) for \(\beta\) in some indexing set \(J\text{.}\) Prove or disprove each of the following. If a statement is not true, is either containment true? Prove your answers.
Let \(A\) and \(B\) be nonempty sets, and let \(f: A \to B\) be a bijection. Prove that
(a)
For every \(x\) in \(A\text{,}\)\(\left( f^{-1} \circ f \right)(x) = x\text{.}\)
(b)
For every \(y\) in \(B\text{,}\)\(\left( f \circ f^{-1}\right)(y) = y\text{.}\)
9.
Let \(R\text{,}\)\(S\text{,}\) and \(T\) be sets, and let \(g: R \to S\) and \(h : S \to T\) be functions. Let \(O\) be a subset of \(T\text{.}\) Show that \((h \circ g)^{-1}(O) = g^{-1}(h^{-1}(O)\text{.}\)
10.
Let \(X_1\) and \(X_2\) be nonempty sets, and let \(X = X_1 \times X_2\text{.}\) Define \(\pi_i :X \to X_i\) by \(\pi_i(x) = x_i\text{,}\) where \(x = (x_1,x_2)\text{.}\) We call \(\pi_i\) the projection of \(X\) onto \(X_i\text{.}\) Let \(Y_1\) and \(Y_2\) be nonempty sets, and let \(Y = Y_1 \times Y_2\text{.}\) Assume that for each \(i\) there is a function \(f_i : X_i \to Y_i\text{.}\) For example, let \(X_i = \{i,i+1\}\) and \(Y_i = \{-i, -i-1\}\text{.}\) We could then define \(f_i\) by \(f_i(x) = -x\) for \(i\) either \(1\) or \(2\text{.}\)
(a)
Prove that \(\pi_i\) is a surjection for each \(i\text{.}\)
(b)
Prove that there is a unique function \(f: X \to Y\) such that \(\pi_i \circ f = f_i \circ \pi_i\) for each \(i\text{.}\) (Note that one of the \(\pi_i\) maps \(X\) to \(X_i\) and the other maps \(Y\) to \(Y_i\text{.}\))
(c)
The function \(f\) from part (b) is denoted as \(f = f_1 \times f_2\text{.}\) Let \(Z_1\) and \(Z_2\) be two nonempty sets, and let \(Z = Z_1 \times Z_2\text{.}\) Assume that there are functions \(g_i : Y_i \to Z_i\) for each \(i\text{.}\) Show that
Start by calculating several outputs for the function before you attempt to write a proof. In exploring whether or not the function is an injection, it might be a good idea to use cases based on whether the inputs are even or odd. In exploring whether \(f\) is a surjection, consider using cases based on whether the output is positive or less than or equal to zero.
12.
An operation \(*\) on a set \(S\) is a function from \(S \times S\) to \(S\) that assigns to the pair \((x,y) \in S \times S\) the element \(x*y\) in \(S\text{.}\) For example, addition of integers can be defined as a function \(f : \Z \times \Z \to \Z\) that maps the pair \((a,b)\in \Z \times \Z\) to the integer \(f(a,b) = a+b\text{.}\)
(a)
Is the function \(f\) an injection? Justify your conclusion.
(b)
Is the function \(f\) a surjection? Justify your conclusion.
13.
Let \(A\text{,}\)\(B\text{,}\) and \(C\) be sets and let \(f : A \to B\) and \(g : B \to C\) be functions.
(a)
Is it true that if \(g \circ f\) is an injection, then both \(f\) and \(g\) are injections? If the answer is no, are there any conditions that \(f\) or \(g\) must satisfy if \(g \circ f\) an injection? Prove your answers.
(b)
Is it true that if \(g \circ f\) is a surjection, then both \(f\) and \(g\) are surjections? If the answer is no, are there any conditions that \(f\) or \(g\) must satisfy if \(f \circ g\) a surjection? Prove your answers.
14.
(a)
Is composition of functions a commutative operation? Prove your answer.
(b)
Is composition of functions an associative operation? Prove your answer.
15.
(a)
Define \(f: \Z_5 \to \Z_5\) by \(f\left( [x] \right) = \left[x^2 + 4 \right]\) for all \([x] \in \Z_5\text{.}\) Write the inverse of \(f\) as a set of ordered pairs, and explain why \(f^{-1}\) is not a function.
(b)
Define \(g: \Z_5 \to \Z_5\) by \(g\left( [x] \right) = \left[x^3 + 4 \right]\) for all \([x] \in \Z_5\text{.}\) Write the inverse of \(g\) as a set of ordered pairs, and explain why \(g^{-1}\) is a function.
(c)
Is it possible to write a formula for \(g^{-1}\left( [y] \right)\text{,}\) where \([y] \in \Z_5\text{?}\) The answer to this question depends on whether or not it is possible to define a cube root of elements of \(\Z_5\text{.}\) Recall that for a real number \(x\text{,}\) we define the cube root of \(x\) to be the real number \(y\) such that \(y^3 = x\text{.}\) That is, \(y = \sqrt[3]{x}\) if and only if \(y^3 = x\text{.}\) Using this idea, is it possible to define the cube root of each element of \(\Z_5\text{?}\) If so, what is \(\sqrt[3]{[0]}\text{,}\)\(\sqrt[3]{[1]}\text{,}\)\(\sqrt[3]{[2]}\text{,}\)\(\sqrt[3]{[3]}\text{,}\) and \(\sqrt[3]{[4]}\text{.}\)
(d)
Now answer the question posed at the beginning of part (c). If possible, determine a formula for \(g^{-1}\left([y] \right)\) where \(g^{-1}: \Z_5 \to \Z_5\text{.}\)
16.
Let \(A\) be the set of all functions \(f:[a,b] \to \R\) that are continuous on \([a,b]\) (use your memory of continuous functions from calculus for this problem). Let \(B\) be the subset of \(A\) consisting of all functions possessing a continuous derivative on \([a,b]\text{.}\) Let \(C\) be the subset of \(B\) consisting of all functions whose value at \(a\) is 0.
(a)
(i)
Give an example of a function that is in \(A\) and not in \(B\) with \([a,b] = [-1,1]\text{.}\)
(ii)
Give an example of a function that is in \(B\) but not in \(C\) with \([a,b] = [-1,1]\text{.}\)
(iii)
Give an example of a function that is in \(C\) with \([a,b] = [-1,1]\text{.}\)
Show that \(h\) is invertible by finding a function \(g : C \to A\) such that \(g\) and \(h\) are inverse functions.
17.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
If \(A\) is a subset of \(X\text{,}\) then \(A \subseteq f^{-1}(f(A))\text{.}\)
(b)
If \(A\) is a subset of \(X\text{,}\) then \(f^{-1}(f(A)) \subseteq A\text{.}\)
(c)
If \(B\) is a subset of \(Y\text{,}\) then \(B \subseteq f(f^{-1}(B))\text{.}\)
(d)
If \(B\) is a subset of \(Y\text{,}\) then \(f(f^{-1}(B)) \subseteq B\text{.}\)
(e)
If \(A_1\) and \(A_2\) are subsets of \(X\) with \(A_1 \subseteq A_2\text{,}\) then \(f(A_1) \subseteq f(A_2)\text{.}\)
(f)
If \(B_1\) and \(B_2\) are subsets of \(Y\) with \(B_1 \subseteq B_2\text{,}\) then \(f^{-1}(B_1) \subseteq f^{-1}(B_2)\text{.}\)
(g)
If \(B_1\) and \(B_2\) are subsets of \(Y\) with \(B_1 \subseteq B_2\text{,}\) then \(f^{-1}(B_2) \subseteq f^{-1}(B_1)\text{.}\)
(h)
If \(A_1\) and \(A_2\) are subsets of \(X\text{,}\) then \(f(A_1 \cup A_2) = f(A_1) \cup f(A_2)\text{.}\)
(i)
If \(B_1\) and \(B_2\) are subsets of \(Y\text{,}\) then \(f^{-1}(B_1 \cup B_2) = f^{-1}(B_1) \cup f^{-1}(B_2)\text{.}\)
(j)
If \(A_1\) and \(A_2\) are subsets of \(X\text{,}\) then \(f(A_1 \cap A_2) = f(A_1) \cap f(A_2)\text{.}\)
(k)
If \(B_1\) and \(B_2\) are subsets of \(Y\text{,}\) then \(f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)\text{.}\)
(l)
If \(A_1\) and \(A_2\) are subsets of \(X\text{,}\) then \(f(A_1 \setminus A_2) = f(A_1) \setminus f(A_2)\text{.}\)
(m)
If \(B_1\) and \(B_2\) are subsets of \(Y\text{,}\) then \(f^{-1}(B_1 \setminus B_2) = f^{-1}(B_1) \setminus f^{-1}(B_2)\text{.}\)
