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280wk1_x4 - What's It All About Why is it computer science...

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What’s It All About? Continuous mathematics— calculus —considers objects that vary continuously distance from the wall Discrete mathematics considers discrete objects, that come in discrete bundles number of babies: can’t have 1.2 The mathematical techniques for discrete mathematics differ from those for continuous mathematics: counting/combinatorics number theory probability logic We’ll be studying these techniques in this course. 1 Why is it computer science? This is basically a mathematics course: no programming lots of theorems to prove So why is it computer science? Discrete mathematics is the mathematics underlying al- most all of computer science: Designing high-speed networks Finding good algorithms for sorting Doing good web searches Analysis of algorithms Proving algorithms correct 2 This Course We will be focusing on: Tools for discrete mathematics: computational number theory (handouts) * the mathematics behind the RSA cryptosystems a little graph theory (Chapter 3) counting/combinatorics (Chapter 4) probability (Chapter 6) * randomized algorithms for primality testing, rout- ing logic (Chapter 7) * how do you prove a program is correct Tools for proving things: induction (Chapter 2) (to a lesser extent) recursion First, some background you’ll need but may not have . . . 3 Sets You need to be comfortable with set notation: S = { m | 2 m 100 , m is an integer } S is the set of all m such that m is between 2 and 100 and m is an integer. 4
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Important Sets (More notation you need to know and love . . . ) N (occasionally IN ): the nonnegative integers { 0 , 1 , 2 , 3 , . . . } N + : the positive integers { 1 , 2 , 3 , . . . } Z : all integers { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . } Q : the rational numbers { a/b : a, b Z, b 6 = 0 } R: the real numbers Q + , R + : the positive rationals/reals 5 Set Notation • | S | = cardinality of (number of elements in) S ◦ |{ a, b, c }| = 3 Subset : A B if every element of A is an element of B Note: Lots of people (including me, but not the authors of the text) usually write A B only if A is a strict or proper subset of B (i.e., A 6 = B ). I write A B if A = B is possible. Power set: P ( S ) is the set of all subsets of S (some- times denoted 2 S ). E.g., P ( { 1 , 2 , 3 } ) = {∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 }} ◦ |P ( S ) | = 2 | S | 6 Set Operations Union: S T is the set of all elements in S or T S T = { x | x S or x T } ◦ { 1 , 2 , 3 } ∪ { 3 , 4 , 5 } = { 1 , 2 , 3 , 4 , 5 } Intersection: S T is the set of all elements in both S and T S T = { x | x S, x T } ◦ { 1 , 2 , 3 } ∩ { 3 , 4 , 5 } = { 3 } Set Difference: S - T is the set of all elements in S not in T S - T = { x | x S, x / T } ◦ { 3 , 4 , 5 } - { 1 , 2 , 3 } = { 4 , 5 } Complementation: S is the set of elements not in S What is { 1 , 2 , 3 } ?
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