The document summarizes various mathematical concepts from Lewis Carroll's Alice's Adventures in Wonderland chapters 4-6, including: 1) The reflexive property of equality, cause and effect trichotomy, indirect proof, and relative/absolute maximum. 2) Properties of circles, exclusivity of properties, and symmetry property of equality. 3) Use of deductive reasoning by the Cat to conclude it is mad based on how it reacts differently than a dog.

2. Math in Alice in Wonderland Chapters 4-6

3. Reflexive property of equality The property that states that a number is equal to itself. The property that states a = a. http://www.mathwords.com/r/reflexive_property.htm “ The Duchess! The Duchess! Oh my dear paws! Oh my fur and whiskers! She’ll get me executed, as sure as ferrets are ferrets !” Math in A lice in Wonderland

4. Cause and effect trichotomy By this time she had found her way into a tidy little room with a table in the window, and on it (as she had hoped) a fan and two or three pairs of tiny white kid gloves: she took up the fan and a pair of the gloves, and was just going to leave the room, when her eye fell upon a little bottle that stood near the looking-glass. There was no label this time with the words 'DRINK ME,' but nevertheless she uncorked it and put it to her lips. 'I know something interesting is sure to happen ,' she said to herself, 'whenever I eat or drink anything ; so I'll just see what this bottle does. I do hope it'll make me grow large again, for really I'm quite tired of being such a tiny little thing!' Math in A lice in Wonderland

6. Relative Maximum “ There ought to be a book written about me, that there ought! And when I grow up, I’ll write one – but I’m grown up now,” she added in a sorrowful tone; “at least there’s no more room to grow up any more here .” Relative Maximum - The highest point in a particular section of a graph. (a.k.a. Relative Max , Local Max , Local Maximum ) http://www.mathwords.com/l/local_maximum.htm Math in A lice in Wonderland

7. Relative Maximum A B Math in A lice in Wonderland

8. Absolute Maximum “ But then,” thought Alice, shall I never get any older than I am now ? That’ll be a comfort, one way – never to be an old woman – but then – always to have lessons to learn! Oh, I shouldn’t like that!” The highest point over the entire domain of a function or relation. a.k.a. Absolute Max, Global Maximum, Global Max http://www.mathwords.com/l/local_maximum.htm Math in A lice in Wonderland

9. Absolute Maximum A Math in A lice in Wonderland

10. Property of circle/semicircle “ Alice remained looking thoughtfully at the mushroom for a minute, trying to make out which were the two sides of it; and as it was perfectly round, she found this a very difficult question. However, at last she stretched her arms around it as far as they would go, and broke off a bit of the edge with each hand. “ And now which is which?” she said to herself, and nibbled a little of the right-hand bit to try the effect: the next moment she felt a violent blow underneath her chin: it had struck her foot!” Math in A lice in Wonderland

11. Property of circle/semicircle Since a circle has infinitely many lines of symmetry, there are also infinitely many lefts and right. But, the 2 endpoints of a diameter are always on different sides of a circle, so one side is left and the other would be right. Math in A lice in Wonderland

12. Exclusivity of some properties For a minute or two she stood looking at the house, and wondering what to do next, when suddenly a footman in livery came running our of the wood – (she considered him to be a footman because he was in livery: otherwise, judging by his face only, she would have called him a fish)… Next page pls. Math in A lice in Wonderland

13. Exclusivity of some properties “ I can see you’re trying to invent something!” “I – I’m a little girl,” said Alice, rather doubtfully, as she remembered the number of changes she had gone through that day. “A likely story indeed!” said the Pigeon in a tone of the deepest contempt. “I’ve seen a good many little girls in my time, but never one with such a neck as that! No, no! You’re a serpent; and there’s no use denying it. I suppose you’ll be telling me next that you never tasted an egg!” “I have tasted eggs, certainly,” said Alice, who was a very truthful child; “but little girls eat eggs quite as much as serpents do, you know.” “I don’t believe it,” said the Pigeon; “but if they do, why then they’re a kind of serpent, that’s all I can say.” Math in A lice in Wonderland

15. Symmetry property of equality The following property: If a = b then b = a. “ For the Duchess . An invitation from the Queen to play croquet.” The Frog Footman repeated, in the same solemn tone, only changing the order of the words a little, “ From the Queen . An invitation for the Duchess to play croquet.” Math in A lice in Wonderland

16. Symmetry property of equality The following property: If a = b then b = a. X Y = X Y = For the Duchess from the Queen For the Duchess from the Queen Math in A lice in Wonderland

17. *The segment joining 2 points on the same side of a given line will not intersect the given line “ There’s no sort of use in knocking,” said the Footman, “and that for two reasons. First, because I’m on the same side of the door as you are; secondly, because they’re making such a noise inside, no one could possibly hear you.” And certainly there was a most extraordinary noise going on within – a constant howling and sneezing, and every now and then a great crash, as if a dish or kettle had been broken to pieces. Next Math in A lice in Wonderland

18. *The segment joining 2 points on the same side of a given line will not intersect the given line “ Please, then,” said Alice, “how am I to get in?” “There might be some sense in your knocking,” the Footman went on without attending to her, “if we had the door between us. For instance, if you were inside, you might knock, and I could let you out, you know.” Next Math in A lice in Wonderland

19. *The segment joining 2 points on the same side of a given line will not intersect the given line Alice Footman Door Math in A lice in Wonderland

20. Deductive reasoning “ Oh, you can’t help that,” said the Cat: “we’re all mad here. I’m mad. You’re mad.” “How do you know I’m mad?” said Alice. “You must be,” said the Cat, “or you wouldn’t have come here.” Alice didn’t think that proved it at all; however, she went on: “And how do you know that you’re mad?” “ To begin with ,” said the Cat, “ a dog’s not mad. You grant that?” “I suppose so,” said Alice.” “Well, then,” the Cat went on, “you see, a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad .” Math in A lice in Wonderland

21. Real Numbers – “ In that direction,” the Cat said, waving its right paw around, “lives a Hatter: and in that direction” waving the other paw, “lives a March Hare. Visit either you like: they’re both mad.” 0 + - Alice March Hare Hatter Math in A lice in Wonderland

22. Deductive reasoning 1. Assume: A dog’s not mad 2. A dog growls when it’s angry, and wags its tail when it’s pleased. => Property of dogs 3. A cat growls(purrs) when it’s pleased, and wags its tail when it’s angry. 4. The contrapositive of a true statement is always true (If P=>Q, then ~P=>~Q). 5. Therefore, a cat is mad. Math in A lice in Wonderland

23. Thank you Ernest Nathan L. Nogales Eugene Paolo Gabo Neil Astrologo Jethro Daniel Pascasio Charm Mac Group 2