🚧 Setup for reading/practice/additional exercises

chapter_1/1_5/additional_exercises.md

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+# 1.5.8 Additional Exercises
+
+1.
+
+Q: Prove that for any two sets $A$ and $B$, $A \subseteq B$ if and only if
+$A \cup B = B$.
+
+A:
+
+2.
+
+Q: The **intersection** of sets $A$ and $B$, denoted $A \cap B$, is the set of
+all elements that are in both $A$ and $B$.
+
+Prove that for any two sets $A$ and $B$, $A \subseteq B$ if and only if
+$A \cap B = A$.
+
+A:
+
+3.
+
+Q: Prove that for any sets $A$, $B$, and $C$, if $A \cup B \subseteq C$, then
+$A \subseteq C$ and $B \subseteq C$.
+
+A:
+
+4.
+
+Q: Prove that for any sets $A$, $B$, and $C$, if $A \subseteq C$ and
+$B \subseteq C$, then $A \cup B \subseteq C$.
+
+A:
+
+5.
+
+Q: The **difference** of sets $A$ and $B$, written $A$ \ $B$, is the set of all
+elements that are in $A$ but not in $B$.
+
+The **empty set**, written $\emptyset$, is the set that contains no elements.
+
+Prove that if $A$ \ $B = A$ then $A \cap B = \emptyset$.
+
+A:
+
+6.
+
+Q: Prove that if $A$ \ $B = B$ \ $A$ then $A = B$.
+
+A:
+
+7.
+
+Q: Let $f:X \to Y$ be a function, and let $A$ and $B$ be subsets of $X$.
+
+(a) Prove that $f(A \cap B) \subseteq f(A) \cap f(B)$.
+
+(b) Find an example of a function and two sets $A$ and $B$ such that
+$f(A \cap B) \neq f(A) \cap f(B)$.
+
+A:
+
+8.
+
+Q: Let $f: X \to Y$ be a function, and let $A$ and $B$ be subsets of $X$.
+
+(a) Prove that $f(A \cup B) \subseteq f(A) \cup f(B)$.
+
+(b) Prove that $f(A) \cup (B) \subseteq f(A \cup B)$
+
+\(c\) What can you conclude from the two proofs above?
+
+A:
+
+9.
+
+Q: Given a function $f: X \to Y$ and a set $B \subseteq Y$, we define the
+**inverse image** of $B$ under $f$ as the set
+$f^{-1}(B) = \{x \in X: f(x) \in B\}$. That is, it is all the elements in the
+domain that are mapped to elements in $B$.
+
+(A) For $f: \mathbb{N} \to \mathbb{N}$ defined by $f(n) = n^2$, what are each of
+the following sets?
+
+(a) $f^{-1}(\{1, 4, 9\})$
+
+(b) $f^{-1}(\{2, 3, 5, 7\})$
+
+\(c\) $f^{-1}(\{1, 2, \dots, 10\})$
+
+(B) Prove that for any set $C \subseteq X, C \subseteq f^{-1}(f(C))$.
+
+\(C\) Give an example of a function $f$ and a set $C$ such that
+$C \neq f^{-1}*(f(C))$.
+
+(D) Prove that for any set $D \subseteq Y, f(f^{-1}(D)) \subseteq D$.
+
+(E) Give an example of a function $f$ and a set $D$ such that
+$f(f^{-1}(D)) \neq D$.
+
+A:
+
+10.
+
+Q: Let $f: X \to Y$ be a function, and let $A$ and $B$ be subsets of $Y$. Prove
+that $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$.
+
+A:
+
+11.
+
+Q: Let $f: X \to Y$ be a function, and let $A$ and $B$ be subsets of $Y$. Prove
+that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$.
+
+A:
+
+12.
+
+Q: For each relation below, determine whether it is transitive. If it is, prove
+it. If it is not, give a counterexample.
+
+(a) The relation "$|$" (divides) on $\mathbb{Z}$ defined by $a|b$ provided $b$
+is a multiple of $a$.
+
+(b) The relation "$\leq$" (less than or equal to) on $\mathbb{R}$.
+
+\(c\) The relation "$\perp$" (is perpendicular to) on the set of lines in the
+plane.
+
+(d) The relation "$\sim$" (is similar to) on the set of triangles in the plane
+(two triangles are similar if they have the same angles, bu are not necessarily
+the same size).

chapter_1/1_5/definition_1_5_1.md

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+# Definition 1.5.1
+
+A set $A$ is a **subset** of a set $B$, written $A \subseteq B$, provided every
+element of $A$ is also an element of $B$.
+
+The set $B$ is sometimes called a **superset** of $A$.
+
+We say $A$ is a **proper subset** of $B$, written $A \subset B$, provided
+$A \subseteq B$ and $A \neq B$. In other words, if every element in $a$ is an
+element in $B$, and there is at least one element in $B$ that is _not_ in $A$.
+
+## Example 1.5.2
+
+Let $A = \{x \in \mathbb{N}: x < 5\}$ and $B = \{x \in \mathbb{N}: x^2 < 10\}$.
+Is $B \subseteq A$? Is $B$ a _proper_ subset of $A$?
+
+**Solution** We are asking whether every natural number less than $5$ is also a
+natural number whose square is less than $10$. Okay, we could just write out the
+elements of the sets: $A = \{0, 1, 2, 3, 4\}$ and $B = \{0, 1, 2, 3\}$ (since
+$3^2 = 9$ and $4^2 = 16$) . So $B \subseteq A$. But $B \neq A$, so in fact
+$B \subset A$.

chapter_1/1_5/definition_1_5_12.md

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+# Definition 1.5.12
+
+A relation $R$ on a set $A$ is **transitive** provided for all $x$, $y$,
+$z \in A$, if $xRy$ and $yRz$, then $xRz$.
+
+## Example 1.5.13
+
+Consider the relation $~$ on the set of students in your Discrete Math course
+that holds of two students, provided they have some other class together. Is
+this relation transitive?
+
+**Solution** No, not necessarily (although for some sets of students it could
+be). For example, suppose Alice has another class with Bruce, say Introduction
+to Programming. Carlos is not in Intro to Programming, but he and Bruce are both
+in Organic Chemistry. So then Alice $~$ Bruce and Bruce $~$ Carlos, but it might
+nobe the case that Alice $~$ Carlos (since Alice need not be in Organic
+Chemistry with Carlos).

chapter_1/1_5/definition_1_5_15.md

@@ -0,0 +1,5 @@
+# Definition 1.5.15
+
+Let $v$ be a vertex in a graph $G$. The **degree** of $v$, written $d(v)$, is
+the number of edges that contain $v$, i.e., the number of edges **incident** to
+$v$.

chapter_1/1_5/definition_1_5_5.md

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+# Definition 1.5.5
+
+A function $f : A \to B$ is **injective** (or **one-to-one**) provided every
+element in$ B$ is the image of at most one element in $A$. In other words, no
+element in $B$ is the _output_ for more than one _input_ from $A$.

chapter_1/1_5/definition_1_5_9.md

@@ -0,0 +1,16 @@
+# Definition 1.5.9
+
+Given a function $f : X \to Y$ and a set $A \subseteq X$, we define the **image
+of** $A$ **under** $f$ to be the set $f(A) = \{f(a) \in Y: a \in A\}$. That is,
+$f(A)$ is the set of all outputs of the function for inputs in $A$.
+
+## Example 1.5.10
+
+Let $f: \mathbb{N} \to \mathbb{N}$ be defined by $f(n) = 2n$. Let
+$A = \{1, 2, 3\}$. Find $f(A)$.
+
+**Solution**. Evaluate each element of $A$ by $f$.
+
+$$ f(1) = 2; \quad f(2) = 4; f(3) = 6. $$
+
+We want the set of these outputs. So $f(A) = \{2, 4, 6}$.

chapter_1/1_5/practice_problems.md

@@ -0,0 +1,123 @@
+# 1.5.7 Practice Problems
+
+1.
+
+Q: Given sets $A$ and $B$, the **intersection** of $A$ and $B$, written
+$A \cap B$, is the set of all elements that are in both $A$ and $B$.
+
+Suppose you wanted to prove that if $A \cap B = B$ then $B \subseteq A$.
+
+Which would be a good start to this proof if you used a direct proof?
+
+A. Let $a$ be an element of $A \cap B$.
+
+B. Let $b$ be an element of $B$.
+
+C. Let $a$ be an element of $A$.
+
+D. Suppose there is an element $b$ in $B$ that is not in $A$.
+
+A:
+
+2.
+
+Q: Suppose you wanted to prove that for all sets $A$ and $B$ that
+$A \cap B \subseteq A$. Which of the following would be a good start to a proof
+by contradiction?
+
+A. Suppose there is an element $a$ in $A$ that is not in $A \cap B$.
+
+B. Suppose there is an element $a$ in $A \cap B$ that is not in $A$.
+
+C. Let $a$ be an element of $A$.
+
+D. Let $a$ be an element of $A \cap B$.
+
+A:
+
+3.
+
+Q: Arrange some of the statements below to form a correct proof of the following
+statement: "For any sets $A$ and $B$, if $B \subseteq A \cap B$" then
+$B \subseteq A$".
+
+- Therefore $B \subseteq A \cap B$
+
+- Since $A \cap B$ contains all the elements that are in both $A$ and $B$, $b$
+  is an element of $A$.
+
+- Then $b$ is an element of $A \cap B$ since $B \subseteq A \cap B$.
+
+- Let $b$ be an element of $A \cap B$.
+
+- Suppose $B \subseteq A$.
+
+- Suppose $B \subseteq A \cap B$, then let $b$ be an element of $B$.
+
+- Then $b$ is an element of $B$ since $B \subseteq A \cap B$.
+
+- Therefore $B \subseteq A$.
+
+- Suppose $A \subseteq B$.
+
+A:
+
+4.
+
+Q: Prove that for any sets $A$ and $B$, $(A \cap B) \cup A$ = A$. Arrange the
+statements below to form a correct proof.
+
+- So in particular, $x$ is an element of $A$.
+
+- Second, we will prove that $A \subseteq (A \cap B) \cup A$.
+
+- Let $x$ be an element of $(A \cap B) \cup A$.
+
+- Therefore $A \subseteq (A \cap B) \cup A$.
+
+- First we will prove that $(A \cap B) \cup A \subseteq A$.
+
+- Therefore $(A \cap B) \cup A \subseteq A$.
+
+- Since $(A \cap B) \cup A \subseteq A$ and $A \subseteq (A \cap B) \cup A$, we
+  have $(A \cap B) \cup A = A$.
+
+- Then $x$ is an element of $(A \cap B) \cup A$, since $x$ is in $A$ or in the
+  other set.
+
+- Then $x$ is an element of $A \cap B$, or $x$ is an element of $A$.
+
+- Let $x$ be an element of $A$.
+
+A:
+
+5.
+
+Q: Let $f: X \to Y$ be a function and let $B \subseteq Y$ be a subset of the
+codomain. Define the **inverse image** of $B$ under $f$ to be the set
+$f^{-1}(B) = \{x \in X: f(x) \in B\}$. That is, it is all the elements in the
+domain that are mapped to elements in $B$.
+
+Prove that if $B_1 \subseteq B_2$ are subsets of the codomain, then
+$f^{-1}(B_1) \subseteq f^{-1}(B_2)$. Arrange some of the statements below to
+form a correct proof.
+
+- Thus $B_1 \subseteq B_2$ .
+
+- This means that $f(a)$ is an element of $B_1$.
+
+- Therefore $b$ is an element of $B_2$.
+
+- Therefore $f^{-1}(B_1) \subseteq f^{-1}(B_2)$.
+
+- Suppose $B_1 \subseteq B_2$.
+
+- Since $B_1 \subseteq B_2$, $f(a)$ is an element of $B_2$.
+
+- This then means that $a$ is an element of $f^{-1}(B_2)$.
+
+- Let $b$ be an element of $B_1$.
+
+- Let $a$ be an element of $f^{-1}(B_1)$.
+
+A:

chapter_1/1_5/preview_activity.md

@@ -14,31 +14,80 @@ in $A$ is also an element of $B$.
 b. Given sets $A$ and $B$, the union of $A$ and $B$, written $A \cup B$, is the
 set containing every element that is in $A$ and $B$ or both.
 
-(a) Let $B = \{1, 3, 5, 7, 9\}$. Give an example of a set $A$ containing 3
+(a)
+
+Q: Let$B = \{1, 3, 5, 7, 9\}$. Give an example of a set $A$ containing 3
 elements that is a subset of $B$.
 
 What is $A \cup B$ for a set $A$ you gave as an example?
 
-(b) Give an example of two distinct sets $A$ and $B$ such that $A \cup B = B$.
+A:
+
+$$ A = \{1, 3, 5\} $$
+
+$$ A \cup B = \{1, 3, 5, 7, 9\} $$
+
+(b)
+
+Q: Give an example of two distinct sets $A$ and $B$ such that $A \cup B = B$.
 
 For the example you gave, is $A \subseteq B$?
 
-\(c\) Find examples, if they exist, of sets $A$ and $B$ such that
+A:
+
+$$ A = \{1, 2\}, \quad B = \{1, 2, 3\} $$
+
+Yes, for this example, $A \subseteq B$.
+
+\(c\)
+
+Q: Find examples, if they exist, of sets $A$ and $B$ such that
 $A \cup B \neq B$.
 
 For the example you gave, is $A \subseteq B$?
 
+A:
+
+$$ A = \{1, 2, 3\} \quad B = \{1, 2\} $$
+
+$$ A \cup B = \{1, 2, 3\} $$
+
+In this case $A$ is not a subset of $B$.
+
 2.
 
 Which of the following are always true?
 
-A. For any sets $A$ and $B$, $A \cup B \subseteq B$.
+A.
 
-B. For any sets $A$ and $B$, $B \subseteq A \cup B$.
+Q: For any sets $A$ and $B$, $A \cup B \subseteq B$.
 
-C. For any sets $A$ and $B$, if $A \subseteq B$, then $A \cup B \subseteq B$.
+A: No, we just demonstrated that this was not true in 1a by showing example sets
+this is not the case.
 
-D. For any sets $A$ and $B$, if $A \cup B = B$, then $A \subseteq B$.
+B.
+
+Q: For any sets $A$ and $B$, $B \subseteq A \cup B$.
+
+A: Yes this is always true. Since $A \cup B$ will always contain elements from
+both $A$ and $B$, this means that every element of $B$ is automatically in
+$A \cup B$.
+
+C.
+
+Q: For any sets $A$ and $B$, if $A \subseteq B$, then $A \cup B \subseteq B$.
+
+A: Yes this is always true. $A \subseteq B$ means that set $B$ contains all
+elements of $A$, and so therefore the union of $A \cup $B$ will always be a
+subset of $B$.
+
+D.
+
+Q: For any sets $A$ and $B$, if $A \cup B = B$, then $A \subseteq B$.
+
+A: Yes, this is always true. If $A \cup B = B$, this means that any element in
+$A$ must also be in $B$, but $A$ must also not contain any elements not in $B$.
+This makes $A$ a subset of $B$.
 
 3.
 
@@ -50,10 +99,45 @@ $f(A) = \{f(x) : x \in A\}$.
 For the following tasks, let's explore the function
 $f : \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x^2 - 3x + 8$.
 
-(a) Let $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$. Find $f(A)$ ad $f(B)$. Then
-find $f(A) \cup f(B)$.
+(a)
+
+Q: Let $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$. Find $f(A)$ ad $f(B)$. Then find
+$f(A) \cup f(B)$.
+
+A:
+
+$$ f(A) = \{6, 8 \} $$
+
+$$ f(B) = \{12, 26\}  $$
+
+$$ f(A) \cup f(B) = \{6, 8, 12, 26\} $$
 
 (b) Now find $A \cup B$ and $f(A \cup B)$.
 
-\(c\) Give an example, if one exists, of two distinct sets $A$ and $B$ such that
+$$ A \cup B = \{1, 2, 3, 4, 6\} $$
+
+$$ f(A \cup B) = \{6, 8, 12, 26\} $$
+
+\(c\)
+
+Q: Give an example, if one exists, of two distinct sets $A$ and $B$ such that
 $A \subseteq B$ and $f(A) \subseteq f(B)$.
+
+A:
+
+$$ A = \{1, 2\} \quad B = \{1, 2, 3\} $$
+
+$$ A \subseteq B $$
+
+$$ f(A) = \{6\} $$
+
+$$ f(B) = \{6, 8\} $$
+
+$$ f(A) \subseteq f(B) $$
+
+Q: Give an example, if one exists, of two distinct sets $A$ and $B$ such that
+$A \subseteq B$ but $f(A) \nsubseteq f(B)$.
+
+A:
+
+The requested example cannot exist.

chapter_1/1_5/proposition_1_5_11.md

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+# Proposition 1.5.11
+
+Let $f: X \to Y$ be a function, and let $A$ and $B$ be subsets of $X$. If
+$A \subseteq B$, then $f(A) \subseteq f(B)$.

chapter_1/1_5/proposition_1_5_17.md

@@ -0,0 +1,3 @@
+# Proposition 1.5.17
+
+In any graph, the number of vertices with odd degree must be even.

chapter_1/1_5/proposition_1_5_3.md

@@ -0,0 +1,4 @@
+# Proposition 1.5.3
+
+For any sets $A$, $B$, and $C$, if $A \subseteq B$ and $B \subseteq C$, then
+$A \subseteq C$.

chapter_1/1_5/proposition_1_5_7.md

@@ -0,0 +1,4 @@
+# Proposition 1.5.7
+
+Suppose $f : A \to B$ is a function with $A$ and $B$ both finite sets. If
+$|A| > |B|$, then $f$ is not injective.

chapter_1/1_5/reading_questions.md

@@ -0,0 +1,30 @@
+# 1.5.6 Reading Questions
+
+1.
+
+Q: Which of the following is the definition of a function $f: A \to B$ being
+injective?
+
+A. Every element of $B$ is the image of at most one element of $A$.
+
+B. The domain $A$ is a larger set than the codomain $B$.
+
+C. Every element of $A$ is sent to at most one element of $B$.
+
+D. The codomain $B$ is no smaller than the domain $A$.
+
+A:
+
+2.
+
+Q: When would you most likely use element chasing as part of a proof?
+
+A. When proving that one set is a subset of another.
+
+B. When proving that a function is injective.
+
+C. When proving that a relation is transitive.
+
+D. When proving that a graph has an odd number of edges.
+
+A: