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Handouts - CSE 1400 Applied Discrete Mathematics Handouts...

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CSE 1400 Applied Discrete Mathematics Handouts Department of Computer Sciences College of Engineering Florida Tech Spring 2011 Abstract This file contains handouts covering topics in the class. Numbers Numeral Systems Conversion Modular Numbers Sets Boolean Logic Predicate Logic Relations Graphs Equivalences Orders Functions Polynomials Logarithms & Exponentials Integer Functions Permutations Induction Sequences Recurrences Definitions Theorems Proofs Algorithms Problems

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CSE 1400 Applied Discrete Mathematics Numbers Department of Computer Sciences College of Engineering Florida Tech Spring 2011 1 Numbers 1 1 . 1 Natural Numbers 1 1 . 2 Integers 2 1 . 3 Integers Mod n 3 1 . 4 Rational Numbers 3 1 . 5 Floating Point Numbers 5 1 . 6 The Real Numbers 6 1 . 7 Complex Numbers 6 2 Problems on Numbers 7 Abstract Numbers are used for many purposes, chiefly to count discrete things 1 Numbers There are many types of numbers: whole numbers, positive and negative numbers, fractional numbers, and continuous numbers. 1 . 1 Natural Numbers The symbol N is used to refer to the set of natural numbers: zero, one, two, three, . . . . To count things, the natural numbers are used. N = { 0, 1, 2, 3, . . . } The natural numbers are unsigned, that is, no plus ( + ) or minus ( - ) sign is placed in front of a natural number. The natural numbers are closed under addition and multiplication. Addition closure: If n and m are natural numbers, then n + m is a natural number. Multiplication closure: If n and m are natural numbers, then nm is a natural number.
cse 1400 applied discrete mathematics numbers 2 Subtraction and division can be computed on some pairs of natural numbers, but the natural numbers are not closed under these opera- tions. 8 - 9 N ; 5 ÷ 2 N . • If n is less than or equal to m , then the difference m - n is a natural number. • If m is a multiple of n , say m = nk for some natural number k , then the quotient m / n = k is a natural number. A relation < is strict if a < b , then b < a . A relation < is total if for any a and b , either a < b , b < a , or a = b . The natural numbers can be placed in order. 0 < 1 < 2 < 3 < 4 < · · · Less than is a relation on the natural numbers. Less than is strict and total. 1 . 2 Integers Integers are used to increment and decrement counts of things . The integers are the numbers in the set Z = { 0, ± 1, ± 2, ± 3, . . . } The integers are closed under addition, multiplication, and subtrac- The symbol Z is commonly used to refer to the set of integers: zero, plus or minus one, plus or minus two, plus or minus three, . . . . tion. Subtraction closure: If n and m are integers, then m - n is an integer. The integers are signed numbers: They are stored in computer mem- ory with an explicit plus (+) or minus ( - ) sign. Surprisingly, the natural numbers and the integers can be put into a one-to-one correspondence, which mathematicians understand to mean the sets have an equal number of members. A function that establishes this one-to-one correspondence maps the even natural

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Handouts - CSE 1400 Applied Discrete Mathematics Handouts...

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