583652398-Variables-The-Language-of-Sets-The-Language-of-Relations-and-Functions.pptx

Math in the Modern World
GEN ED 4
Variables, The Language of Sets, The
Language of Relations and Functions
with: Mam Kenn

GEN ED 4 Mathematics in the Modern World
Object
ives:
At the end of the lesson, the learners will be able to:
• Discuss the language, symbols, and conventions of mathematics .
• Explain the nature of mathematics as a language.
• Perform operations of mathematical expressions correctly.
• Acknowledge that mathematics is a useful language.

Variables
It is represented by a letter like x or y. A symbol for a value we don’t
know yet.
What is the advantage of
using variables
• It allows you to give temporary name to what you are seeking so that you
can perform concrete computations with it to help discover its possible values.
Examp
le:
• 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 property that 2
2 3 "?
x x
 

“Is there a number with the property that
2
2 3 "?
 
Another way:
• To illustrate the second use of variables, consider the statement:
No matter what number might be chosen, if it is greater than 2, then its
square is greater than 4.

• Introducing a variable to give a temporary name to the number you might
choose enables you to maintain the generality of the statement.
“Is there a number x with the property that 2
2 3 "?
x x
 
No matter what number might be chosen, if it is
greater than 2, then its square is greater than 4.
No matter what number x might be chosen, if x is greater than 2, then is
greater than 4.

Writing Sentences Using
Variables
• Use variables to rewrite the following sentences more formally.
a. Are there numbers with the property that the sum of their squares equals the
square of their sum?
Examp
les:
Soluti
on:
• Are there numbers x and y with the property  
2
2 2 .
x y x y
  
• Are there numbers x and y such that  
2
2 2 .
x y x y
  
• Do there exist any numbers x and y such that  
2
2 2 .
x y x y
  

GED 141 Mathematics in the Modern World
b. Give any real number, its square is non-negative.
Soluti
on:
• Given any real number is non-negative.
2
r, r
• For any real number 2
r, 0.
r 
• For all real number 2
r, 0.
r 
Some Important Kinds of
Mathematical Statements
1. Universal Statement – says that a certain property is true for all elements in a
set. “For all”
Examp
le:
All positive numbers are greater than zero.

GED 141 Mathematics in the Modern World
2. Conditional Statement – says if one thing is true then some other thing also has
to be true. “If – then”
Examp
le:
If 378 is divisible by 18, then 378 is divisible by 6.
3. Existential Statement – says that there is at least one thing for which the
property is true.
Examp
le:
There is a prime number that is even
4. Universal Conditional Statement – a statement that is both universal and
conditional.
Examp
le:
For all animals a, if a is a dog, then a is a mammal.

Examp
les:
• They can be re-written in ways that make them appear to be purely
universal or purely conditional.
• If a is a dog, then a is a mammal.
• If an animal is a dog, then the animal is a mammal.
• For all dogs a, a is a mammal.
• All dogs are mammals.
Rewriting a Universal Conditional
Statement
Direction: Fill in the blanks to rewrite the following statement:
For all real number x, if x is nonzero then is positive.

1. If a real number is nonzero, then its square _______.
positive
2. For all nonzero real number x, ___________.
is positive
3. If x _________________________, then ____________.
𝑖𝑠𝑎𝑛𝑜𝑛𝑧𝑒𝑟𝑜𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟 positive
4. The square of any nonzero real number is _______.
positive
5. All nonzero real number have ______________.
positive squares
What is Universal Existential
Statements
• A statement that is universal because its first part says that a certain
property is true for all objects of a given type, and it is existential because
its second part asserts the existence of something.

• All real numbers have additive inverses.
Examp
les:
Every real number has an additive inverse.
• For all real numbers r, there is an additive inverse for r.
• For all real numbers r, there is a real s, such that s is an additive
inverse .
Rewriting a Universal Existential
Statement
Every pot has a lid.
1. All pots _______.
has a lid
2. For all pots P, there is ________.
a lid for P
3. For all pots P, there is a lid L such that ____________.
L is a lid for P

What is Existential Universal
Statements
• A statement that is existential because its first part asserts that a certain
object exists and is universal because its second part says that the
object satisfies a certain property for all things of a certain kind.
Examp
les:
There is a positive integer that is less than or equal to every
positive integer.
Rewriting a Existential Universal
Statement
There is a person in my class who is at least as old as every
person in my class.

1. Some _______________ is at least as old as _____________________ .
person in my class every person in my class
2. There is a person p in my class that p is __________________________________.
at least as old as every person in my class
3. There is a person p in my class with the property that for every person q in my
class, p is ________________.
at least as old as q
The Language of Sets
A set is a well-defined collection of distinct objectives.
• It usually represented by capital letters.
• The objects of a set are separated by commas.
• The objectives that belong in a set are the elements, or members of the set.

• It can be represented by listing its elements between braces.
• A set is said to be well – defined if the elements in a set are specifically listed.
Examp
les:
A = {a, e, I, o, u} B = {set of plane figures} C = {Ca, Au, Ag}
Notatio
n
• If S is a set, the notation.
► x Є S means that x is an element of S.
► x S means that x is not element of S.


• A variation of notation is used to describe a very large set.
► { 1, 2, 3, …, 100} refer to set of all integers from 1 to 100.
► { 1, 2, 3, …} refer to the set of all positive integers.
• A symbol … is called ellipses and is read “and so forth”.
Using the Set –Roster
Notation
• A set may be specified using the set – roster notation by writing all elements
between braces .
Examp
les:
• Let A = {1,2,3,}, B = {3,2,1}, and C = {1,1,2,2,3,3,3}. What are the element of A, B,
and C? How are A, B, and C related?

Examp
le:
1. Let A = {1,2,3,}, B = {3,2,1}, and C = {1,1,2,2,3,3,3}. What are the element of A,
B, and C? How are A, B, and C related?
A,B, and C have exactly the same three elements, 1,2, and 3. Therefore
A,B, and C are simply represented in different ways. .
• Is {0} = 0?
{0} ≠ 0 because {0} is a set with one element, namely 0, whereas
0 is just a symbol that represents the number. .
2. How many elements are in the set {1,{1}}?
The set {1,{1}} has two elements: 1 and the set whose only element is 1.
.

Examp
le:
3. For each non-negative integer n, let 1 2 0
, ,and .
{ , }.Find U U
U n n U
n  
{1, 1}, {2, 2}, {0, 0} {0,0} {0}.
1 2 0
U U U
       
Cartesian Sets of
Numbers
Some important sets are the following:
1. N = {1, 2, 3, …} = the set of natural numbers.
2. W = {0, 1, 2, 3, …} = the set of whole numbers.
3. Z = {-3, -2, -1, 0, 1, 2, 3,…} = the set of integers.
4. Q = the set of rational numbers (non terminating, non repeating decimals).
5. Q’ = the set of irrational numbers (terminating, non repeating decimals).

6. R = the set of real numbers
7. C = the set of complex numbers
Set –Builder
Notation
• Let S denote and let P(x) be a property that elements of S may or may not
satisfy. We define a new set to be the set of all elements x in S such that P(x) is
true. We denote this set as follows:
 
 
x S P x

the set of all such that

Using the Set –Builder
Notation
• Given that R the set of real numbers, Z the set of all integers, and Z+ the set of
all positive integers, describe the following sets:
 
) 2 5
a x x
   

Soluti
on: 2 5
x x
 
 
 
   


 
) 2 5
b x Z x
   
 
) 2 5
c x Z x

   
Soluti
on:  
 
 
2 5 is the set of all integers
strictly between 2 and 5. It is equal to the
set 1,0,1,2,3,4
x Z x
   


Soluti
on:
 
Since all the integers in Z are positive,
2 5 1,2,3,4
x Z x
 
 
 


    

Definition
Regarding Sets
• A set a finite if the number of elements is countable.
Examp
les:
A = {even numbers less than 10}
B = {days in a week}
• A set a infinite if the numbers of elements cannot be counted.
Examp
les:
A = {even numbers greater than 20}
B = {odd numbers}
C = {stars in the sky}

Equal and Equivalent
Sets
Equal sets are set with exactly the same elements and cardinality.
Examp
les:
A = {c, a, r, e}
B = {r, a, c, e}
Equivalent sets are set with the same number of elements or cardinality.
Examp
les:
A = {a, e, i, o, u}
B = {1, 2, 3, 4, 5}

Joint and Disjoint
Sets
Joint sets are set with common elements (intersection).
Examp
les:
A = {c, a, r, e}
B = {b, e, a, r, s}
Disjoint sets are set with no common elements.
Examp
les:
The set A = {a, b, c} and B = {e, f, g} are disjoint sets, since no elements is common.

The Language of Relations and
Functions
Relations abound in daily life. People are related to each other in many ways as
parents and children, teachers, students, employers, employees, and many others.
In business things that are bought are related to their cost and the amount paid is
related to the number of things bought.
• A relation is a rule that relates values from a set of values (called the domain) to
a second set of values (called the range).
• The elements of the domain can be imagined as input to a machine that applies a
rule to these inputs to generate one or more outputs.

• A relation is also a set of ordered pair (x, y).
Examp
le:
R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
A Relation as a
Subset
Let A = {1, 2} and B = {1, 2, 3} and define a relation R from A to B as follows: Given any
   
, , , meansthat is aninteger.
2
x y
x y AxB x y R 
 
1. State explicitly which ordered pairs are in A x B and which are in R.
2. Is 1 relate to 3? Is 2 relate to 3? Is 2 relate to 2?.
3. What are the domain and range of R?

   
2
x y
x y AxB x y R 
 
1. State explicitly which ordered pairs are in A x B and which are in R.
A x B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}.
  1 1 0
1,1 because 0
2 2
R 
  
  1 2 1
1,2 because
2 2
R  
 
  1 3 2
1,3 because 1
2 2
R  
  
  2 1 1
2,1 because
2 2
R 
 
  2 2 0
2, 2 because 0
2 2
R 
  
  2 3 1
2, 3 because
2 2
R  
 
Thus, R = {(1, 1), (1, 3), (2, 2)}.

   
2
x y
x y AxB x y R 
 
2. Is 1 relate to 3? Is 2 relate to 3? Is 2 relate to 2?.
  1 1 0
1,1 because 0
2 2
R 
  
  1 2 1
1,2 because
2 2
R  
 
  1 3 2
1,3 because 1
2 2
R  
  
  2 1 1
2,1 because
2 2
R 
 
  2 2 0
2, 2 because 0
2 2
R 
  
  2 3 1
2, 3 because
2 2
R  
 
Yes, 1 relate 3 because (1,3) R.
No, 2 relate 3 because (2,3) R.
Yes, 2 relate 2 because (2,2) R.

   
2
x y
x y AxB x y R 
 
3. What are the domain and range of R?
The domain of R is (1, 2) and the range is { 1, 2, 3}.
Thank You and
God Bless!

583652398-Variables-The-Language-of-Sets-The-Language-of-Relations-and-Functions.pptx

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