Lecture1.rtf - Lecture 1 Sets Subset A c B Proper subset A c B How many subsets are there are of S ={a b c d is a E S Can a set contain itself Some say

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Lecture 1 Sets: Subset A c B Proper subset A c B How many subsets are there are of S = {a, b, c, d}? is a E S? Can a set contain itself? - Some say yes, some say no. - testing in python, can you append a list to itself? - Lecture 2 Proof: Proving sqrt(2) is irrational - proving we cannot represent sqrt(2) as a fraction of two whole numbers - to make the proof clearer lets prove some simple theorems (lemmas) first: - lemmas 1: if p^2 is even, then p is even - assume p is odd, since p^2 = p*p, and the product of two odd numbers in odd. Then if p^2 is even, then p must be even - sqrt(2) = p/q - 2 = p^2/q^2 - 2*q^2 = p^2 - since 2*q^2 then p^2 is even, and according to lemmas p is even - letting p = 2*k - 2*q^2 = 4*k^2 - q^2 = 2*k^2 - so q is also even, and the pattern continues infinitely Cartesian product: A X B = {(a, b) | a E A n b E B} Example: let A = {a, b} and B = {

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Unformatted text preview: Functions:- A function F from a set A to a set B- for every element x in set A, there is an element y in set B such that (x, y) E (is an element in) F and:- For all x in A and y and z in B if (x, y) Graphs- a visualization of a relation or kinds of relations. dots represent objects (as in set theory or predicate calculation), edges (links) show relationships between of among objects. Polignac’s conjecture- Every odd number greater than 1 can be expressed as a sum of a power of 2 and a prime number 3 = 2^0 + 2 5 = 261 + 3 7 = 2^2 + 3 9 = 2^2 + 5 Why can’t we prove if it’s true, but is it true?- Begin with a positive integer, If the number is odd multiply by three and add 1, if the number is even divide by 2. You will eventually end up with 1 ex. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1...
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