Speaking Mathematically 1n2z5f

Speaking Mathematically Winter 2013 COMP 1380 Discrete Structures I Computing Science Thompson Rivers University

Course Outline         

Speaking Mathematically Number Systems Computer Arithmetic Logic and Truth Tables Boolean Algebra and Logic Gates Vectors and Matrices Sets and Counting Probability Theory and Distributions Statistics and Random Variables

TRU-COMP1380

Speaking Mathematically

2

Unit Objectives     

Introduce variables for a given sentence. Use of the symbols, , , , in mathematical sentences. Tell if a given list of something is a set. Give the Cartesian product of two sets. Tell if a given relation is a function.

TRU-COMP1380

Speaking Mathematically

3

Unit Contents 1. 2. 3.

Variables The Language of Sets The Language of Ralations and Functions

TRU-COMP1380

Speaking Mathematically

4

1. Variables 

Definition of a variable 



Something that can have a value but you do not know what the value is.

How to introduce variables for a given sentence? 

Example 1 





Example 2 





No matter what number might be chosen, if it is greater than 2, then its square is greater than 4. -> ???

Example 3 



Is there a number with the following property: doubling it and adding 3 gives the same result as squaring it? -> Is there a number x with the following property: 2 x + 3 = x2?

Are there two numbers with the property that the sum of their squares equals the square of their sum?

Example 4 

TRU-COMP1380

Given any real number, its square is nonnegative. Speaking Mathematically

5

Some Important Kinds of Math Statements 

Conditional statement 



Universal conditional statement 



If ..., then ... For all [] ..., if ..., then ...

Universal existential statement  

For all [] ..., there is [] ... E.g., Every real number has an additive inverse. -> For all real number r, there is an additive inverse for r. ->  real number r,  -r  r + (-r) = 0.



Existential universal statement 



There is [] ... such that [] ... E.g., Some positive integer is less than or equal to every positive integer. -> There is a positive integer such that the number is less than or equal to every positive integer. ->  a positive integer m   positive integer n, m  n

TRU-COMP1380

Speaking Mathematically

6

2. The Language of Sets 

Definition of a set: A collection of elements of the same type



Interesting examples 

   



Let A = {1, 2, 3}, B = {3, 1, 2} and C = {1, 1, 2, 3, 3, 3}. How are A, B, and C related? Is {0} = 0? Is {1, {1}} a set? Is {1, a, b} a set? For each nonnegative integer n, let Un = {n, -n}. Find U1, U2, and U0.

An interesting notation 



{x  S | P(x)} meaning ??? the set of elements in S that satisfy P. E.g.,  

TRU-COMP1380

{x  Z | -2 < x < 5} = { ??? } {x  N | -2 < x < 5} = { ??? } Speaking Mathematically

7



Subsets  



AB AB

means x  A, x  B means ??? (x  A, x  B) and ( y  B  y  A)

Interesting example Which of the following are true statements? a) b) c) d)

e) f) g)

2  {1, 2, 3} {2}  {1, 2, 3} 2  {1, 2, 3} {2}  {1, 2, 3} {3, 1, 2}  {1, 2, 3} {2}  {{1}, {2}} {2}  {{1}, {2}}

TRU-COMP1380

Speaking Mathematically

8



Definition of Cartesian product of A and B: A × B = {(a, b) | a  A and b  B}



Is A × B equal to B × A?



Examples Let A = {1, 2, 3} and B = {1, 2} a) b) c)

Find A × B Find B × A Find B × B

TRU-COMP1380

Speaking Mathematically

9

3. The Language of Relations and Functions 

Definition of a relation: Let A and B are sets. A relation (or also called mapping) R from A to B is a subset of A × B. A is called the domain of R and B is called its co-domain.



Example Let A = {1, 2} and B = {1, 2, 3} and define a relation R from A to B as follows: Given any (x, y)  A × B, x R y means that (x – y) / 2 is an integer. a) b) c)

Can you give a diagram for R? Is 1 R 3? Is 2 R 3? Is 2 R 2? What are the domain and co-domain of R?

TRU-COMP1380

Speaking Mathematically

10



1. 2.



Definition of a function: A funcfion F from a set A to a set B is a relation with domain A and co-domain B that satisfies the following two properties: x  A,  y  B  x F y another popular notation F(x) = y x  A and  y and z  B, x F y and x F z -> y = z (What does this mean?) {y  B | x F y for some x  A} is called the image of F, denoted by F(A). Example Let A = {2, 4, 6} and B = { 1, 3, 5}. Which of the relations R, S and T defined below are functions from A to B? a) b) c)

R = {(2, 5}), (4, 1), (4, 3), (6, 5)}  (x, y)  A × B, (x, y)  S means that y = x + 1. 2 T 5, 4 T 1, 6 T 1 (i.e., T(2) = 5; T(4) = 1; T(6) = 1)

TRU-COMP1380

Speaking Mathematically

11



A) F(1) = A; F(2) = B; F(3) = D 1 2 3



A B C D

B) A B C D

C)

A B C D

1 2 3

A B C D

F) 1 2 3

TRU-COMP1380

1 2 3 E)

1 2 3 

D)

A B C D Speaking Mathematically

1 2 3

A B C D 12