# Chapter 1

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McGill University

Mathematics & Statistics (Sci)

MATH 111

Nabil Kahouadji

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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