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chapter_0/0_1/investigate_0.md

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+1.
+
+Q:
+
+The most popular mathematician in the world is throwing a party for all of his
+friednds. To kick things off, they decide that everyone should shake hands.
+Assuming all 10 people at the party each shake hands with every other person
+(but not themselves, obviously) exactly once, how many handshakes take place?
+
+1A:
+
+If we pair off the people into sets, then we get something like this:
+
+{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, {1, 10}
+
+{2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2, 8}, {2, 9}, {2, 10}
+
+{3, 4}, {3, 5}, {3, 6}, {3, 7}, {3, 8}, {3, 9}, {3, 10}
+
+{4, 5}, {4, 6}, {4, 7}, {4, 8}, {4, 9}, {4, 10}
+
+{5, 6}, {5, 7}, {5, 8}, {5, 9}, {5, 10}
+
+{6, 7}, {6, 8}, {6, 9}, {6, 10}
+
+{7, 8}, {7, 9}, {7, 10}
+
+{8, 9}, {8, 10}
+
+{9, 10}
+
+As you can see this just becomes 10+9+8+7+6+5+4+3+2+1=55 handshakes. This likely
+is alluding to basic summation mathematical induction.
+
+2.
+
+Q: At the warm-up event for Oscar's All-Star Hot Dog Eating Contest, Al ate one
+hot dog. Bob then showed him up by eating three hot dogs. Not to be outdone,
+Carl ate five. This continued with each contestant eating two more hot dogs than
+the previous contestant. How many hot dogs did Zeno (the 26th and final
+contestant) eat? How many hot dogs were eaten in total?
+
+A:
+
+This is another summation mathematical induction function, just slightly
+different:
+
+(1) +
+
+(1 + 2) +
+
+(3 + 2) +
+
+(5 + 2) +
+
+(7 + 2) +
+
+(9 + 2) +
+
+(11 + 2) +
+
+(13 + 2) +
+
+(15 + 2) +
+
+(17 + 2) +
+
+(19 + 2) +
+
+(21 + 2) +
+
+(23 + 2) +
+
+(25 + 2) +
+
+(27 + 2) +
+
+(29 + 2) +
+
+(31 + 2) +
+
+(33 + 2) +
+
+(35 + 2) +
+
+(37 + 2) +
+
+(39 + 2) +
+
+(41 + 2) +
+
+(43 + 2) +
+
+(45 + 2) +
+
+(47 + 2) +
+
+(49 + 2) +
+
+(51 + 2) +
+
+(53 + 2) = 55
+
+(1) + (1 + 2) + (3 + 2) + (5 + 2) + (7 + 2) + (9 + 2) + (11 + 2) + (13 + 2) +
+(15 + 2) + (17 + 2) + (19 + 2) + (21 + 2) + (23 + 2) + (25 + 2) + (27 + 2) +
+(29 + 2) + (31 + 2) + (33 + 2) + (35 + 2) + (37 + 2) + (39 + 2) + (41 + 2) +
+(43 + 2) + (45 + 2) + (47 + 2) + (49 + 2) + (51 + 2) + (53 + 2) = 784
+
+55 is how many Zeno ate, and the total hot dogs that were eaten is : 784
+
+3.
+
+Q: After excavating for weeks, you finally arrive at the burial chamber. The
+room is empty except for two large chests. On each is carved a message
+(strangely English):
+
+- Exactly one of these chests contains a treasure, while the other is filled
+  with deadly immortal scorpions.
+
+- For either chest, if the chest's message is true, then the chest contains
+  treasure.
+
+The problem is, you don't know whether the messages are true or false. What do
+you do?
+
+A: Honestly, I'd probably walk away, but since that's likely not the answer,
+we'd likely have to consider that these are relating to a form of truth tables.
+
+The first chest's message claims that one chest contains scorpions, the other
+treasure
+
+So if we say T="contains treasure", and f="contains scorpions", then we can say
+that Chest_1 and Chest_2:
+
+Chest_1 = T Chest_2 = F
+
+The other chest's message claims that if it's own message or the other chest's
+message is true, then the chest contains treasure.
+
+This is a contradiction, because if the second chest's message is true, then
+that chest is the one that contains treasure, and the other chest is the one
+that contains scorpions.
+
+If the second chest's message is false, then the chest doesn't contain treasure,
+but then you don't know if the second chest contains treasure or scorpions,
+because the chest containing treasure is no longer contingent on whether the
+chest's message is true or not.
+
+This is probably best expressed again, in some form, of truth notation.
+
+4.
+
+Q: Back in the days of yore, five small towns decided they wanted to build roads
+directly connecting each pair of towns. While the towns had plenty of money to
+build roads as long and as winding as they wished, it was very important that
+the roads not intersect with each other (as stop signs had not yet been
+invented). Also, tunnels and bridges were not allowed, for moral reasons. It is
+possible for each of these towns to build a road to each of the four other towns
+without creating any intersections?
+
+A: This seems possible, as you could just create two roads from each town to at
+least two other towns (a pair as the question alludes to). But something tells
+me this is a trick question.

chapter_0/0_1/reading_questions.md

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+1. Right now, how would you describe what **discrete** mathematics is about, if
+   you were telling your friends about the class you are in? Write one or two
+   sentences.
+
+Discrete Mathematics is about performing some mathematical logic where the
+inputs and outputs of some function contain only elements that are separate,
+_i.e._ they are discrete. The main problems solved by discrete mathematics deal
+with cominatorics, sequences, symbolic logic, and graph theory, though there are
+others.
+
+2. What questions do you have after reading this section? Write at least one
+   question about the content of this section that you are curious about.
+
+The chest problem interested me as the second chest's message creates a logical
+fallacy, such as "This statement is false." I wonder how this problem would be
+"solved", if it's even possible, with discrete math.

chapter_0/0_2/0_2_8_reading_questions.md

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+1.
+
+Q: Think back to the domino problem, at the beginning of this section. We asked
+how many dominoes are in a double-six domino _set_. Is this really a set, in our
+mathematical sense? What discrete structure would you use to represent each
+domino individually?
+
+A: No, the domino set is _not_ a set, because the order of the number matters so
+you don't repeat any pair. Thusly a sequence, or tuple, would be more
+appropriate to represent each domino individually.
+
+2.
+
+Q: A double-zero domino set would contain only one domino (both sides showing
+0). A double-one set would contain this plus the dominoes (1, 0) and (1, 1). We
+can continue in this way, creating a sequence of domino sets. Find the next
+three terms of this sequence.
+
+1, 3, _, _, _ ...
+
+A: 1, 3, 6, 10, 15
+
+The reason for this is we are summing up all the dominos, you can count starting
+like we wrote out in our investigation:
+
+(0, 6), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)
+
+(0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5)
+
+(0, 4), (1, 4), (2, 4), (3, 4), (4, 4) = 5 tuples
+
+(0, 3), (1, 3), (2, 3), (3, 3) = 4 tuples
+
+(0, 2), (1, 2), (2, 2) = 3 tuples
+
+(0, 1), (1, 1) = 2 tuples
+
+(0, 0) = 1 tuples
+
+1, (1 + 2), (1 + 2 + 3), (1 + 2 + 3 + 4), (1 + 2 + 3 + 4 + 5)
+
+3.
+
+Q: What questions do you have after reading this section? Write at least one
+question about the content of this section that you are curious about.
+
+A: The author makes not of graphs with edges and vertices that are related by
+some function. In the example given, there are graphs where edges are made based
+off of summing up to 7, or in the other, based off those same sets are connected
+by edges based off of if their sum is even. I don't really have a question, but
+the logic based off of these relationships are established. In other words, how
+the cardinality is established and why.

chapter_0/0_2/investigate_0.md

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+2.0:
+
+Q: A double-six domino set consists of tiles containing paris of numbers, each
+from 0 to 6. How many tiles are in a double-six domino set? How many dominoes
+are in a double-nine domino set? How many dominoes are in a double-$n$ domino
+set?
+
+This is a combinatorics problem. It is a summation.
+
+ON a double-six domino set, we can think of each domino being like a bunch of
+tuples:
+
+(0, 6), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)
+
+(0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5)
+
+(0, 4), (1, 4), (2, 4), (3, 4), (4, 4)
+
+(0, 3), (1, 3), (2, 3), (3, 3)
+
+(0, 2), (1, 2), (2, 2)
+
+(0, 1), (1, 1)
+
+(0, 0)
+
+A: The author makes not of graphs with edges and vertices that are related by
+some function. In the example given, there are graphs where edges are made based
+off of summing up to 7, or in the other, based off those same sets are connected
+by edges based off of if their sum is even. I don't really have a question, but
+the logic based off of these relationships are established. In other words, how
+the cardinality is established and why.
+
+And on a double-nine:
+
+(0, 9), (1, 9), (2, 9)