🚧 Setup for 1.5

chapter_1/1_5/investigate.md

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+# Investigate!
+
+Q: Suppose there are 15 people at a party. Most people know each other already,
+but there are still some people who decide to shake hands. Is it possible for
+everyone at the party to shake hands with exactly three other people?
+
+A:
+
+Let $n$ be the amount of people at the party. Since every person must shake
+hands with exactly 3 people, then we can represent the amount of handshakes that
+takes place at the party to be $3n$.
+
+To determine if we have enough people at the party where everyone gets to shake
+the hand of exactly three other people, we can utilize what is known as the
+[Pigeonhole Principle](https://en.wikipedia.org/wiki/Pigeonhole_principle) where
+instead of pigeons we consider the amount of handshakes $3n$ and then evaluate
+the remainder of how many containers, or "handshake contributions" every
+handshake creates, in this case $2$.
+
+To understand this, consider that every time a single person shakes another
+person's hands, it counts as 2 handshakes:
+
+Alice shakes Bob's hand, but Bob also shakes Alice's hands. For the context of
+this problem, 2 handshakes occur every time two individuals shake hands.
+
+If we tally up the total amount of handshakes that must take place at this
+party:
+
+$$ 3n =  3 * 15 = 45 \text{ handshake contributions} $$
+
+And we look at the remainder of how many handshakes are contributed:
+
+$$ 45 \not\equiv 0 \mod 2$$
+
+We can see that we'll always need an even amount of people at this party for
+everyone to be able to shake the hands of exactly 3 people.

chapter_1/1_5/preview_activity.md

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+# Preview Activity
+
+In this preview activity, we will explore some basic properties of sets and
+functions. Later in this section, we will write proofs about these ideas.
+
+1.
+
+Remember that a set is just a collection of elements. Here are two definitions
+about sets:
+
+a. A set $A$ is a subset of $B$, written $A \subseteq B$, provided every element
+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
+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 $A \cup B = B$.
+
+For the example you gave, is $A \subseteq B$?
+
+\(c\) Find examples, if they exist, of sets $A$ and $B$ such that
+$A \cup B \neq B$.
+
+For the example you gave, is $A \subseteq B$?
+
+2.
+
+Which of the following are always true?
+
+A. For any sets $A$ and $B$, $A \cup B \subseteq B$.
+
+B. For any sets $A$ and $B$, $B \subseteq A \cup B$.
+
+C. For any sets $A$ and $B$, if $A \subseteq B$, then $A \cup B \subseteq B$.
+
+D. For any sets $A$ and $B$, if $A \cup B = B$, then $A \subseteq B$.
+
+3.
+
+For any function $f:\mathbb{N} \to \mathbb{N}$ and any set
+$A \subseteq \mathbb{N}$, we can define the image of $A$ under $f$ to be the set
+of all outputs of $f$ when the input is an element of $A$. We write this as
+$f(A) = \{f(x) : x \in A\}$.
+
+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 $f(A) \cup f(B)$.
+
+(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
+$A \subseteq B$ and $f(A) \subseteq f(B)$.

leftoff.txt

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