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MATH 111 (8)

Chapter 1

5 Pages
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School
McGill University
Department
Mathematics & Statistics (Sci)
Course
MATH 111
Professor
Nabil Kahouadji
Semester
Winter

Description
MATH 111 – Winter 2012 1: NUMERATION SYSTEMS AND SETS 1.1 Numeration Systems n power: if a is any number and n is a natural number, then a to the power n is: a = a x a x a x a; any nonzero number a to the power 0 is 1: a = 1 1.1.1 Hindu-Arabic Numeration System - base ten system; decimal numeration system position: 3 2 1 0 number: 2 1 3 4 2134 ten 2 · 10 + 1 · 10 + 3 · 10 + 4 · 100 - unit = 1 - long = 10 - flat = 100 - block = 1000 1.1.2 Other Numeration Base Systems - base five numerals o 5 units makes 1 long (instead of 10 units making 1 long in base ten) o ie/ 12five 7 (1 long plus 2 units) ie/ 234 five 2 flats + 3 longs + 4 units = 69 = 2 · 5 + 3 · 5 + 4 · 5 = 69 ie/ convert 11234 fiveo base ten position 4 3 2 1 0 number 1 1 2 3 4 11234 = 1· 5 + 1 · 5 + 2 · 5 + 3 · 5 + 4 · 5 five = 625 + 125 + 50 + 15 + 4 = 819 ie/ convert 2897 to base five 1. list the powers for desired base 2. stop at first power greater than your number 3. divide your number by highest power not over the number using long division 4. divide until you reach 0 5. numbers across top are your answer 0 1 2 3 4 5 5 = 1 5 = 5 5 = 25 5 = 125 5 = 625 [2897] 5 = 3125 - binary = base two numerals o only uses the numbers 0, 1 MATH 111 – Winter 2012 ie/ convert 110101 two to base ten position: 5 4 3 2 1 0 number: 1 1 0 1 0 1 5 4 3 2 1 0 110101 two= 1 · 2 + 1 · 2 + 0 · 2 + 1 · 2 + 0 · 2 + 1 · 2 = 32 + 16 + 0 + 4 + 0 + 1 = 53 ie/ convert 37 to base two 0 1 2 3 4 5 6 2 = 1 2 = 2 2 = 4 2 = 8 2 = 16 2 = 32 [37] 2 = 64 1.2 Describing Sets 1.2.1 The Language of Sets set: collection of objects; order doesn’t matter; repetition not allowed ie/ A = {a, b, d, c … x, y, z} element: individual items in a set - symbolize element belonging to set: z Є A - symbolize element not belonging to set: è Є A - natural number: = {1, 2, 3, 4, 5…} - 2 ways to describe a set o listing method: X = {1, 2, 3, 4…8} o set builder notation: X = {x | x Є where x < 8} ie/ write the following sets using set builder notation A = {2, 4, 6, 8, 10…} A = {x | x = 2p and p Є } B = {1, 3, 5, 7…} B = {x | x = 2p-1 and p Є } 1.2.2 One-to-One Correspondence equal sets: two sets A and B are equal if, and only if, they contain exactly the same elements; we write then, A = B, otherwise A ≠ B one-to-one correspondence: if the elements of sets P and S can be paired so that for each element of P there is exactly one element of S and for each element of S there is exactly one element of P, then the two sets P and S are said to be in one-to-one correspondence - if two sets have the same number of elements they are in one-to-o
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