Nigger Flashcards
A fundamental dscipline that plays a critical role in various aspects of life, science and technology
Importance of Mathematics
Teaches logical reasoning, critical thinking, and ———.
Problem-Solving Skills
Core concepts of mathematics
Numbers
Operations
Patterns and relationships
The foundation of mathematics, including natural —–
Numbers
Addition, subtraction, multiplication, division, etc
Operations
Identifying and analyzing patterns to understand relationships between numbers and variables
Patters and relationships
Science and engineering:
Example: Calculating the trajectory of a spacecraft using calculus and trigonometry
Application: Engineers use mathematics to design structure, analyze data, and create simulations.
Applications of mathematics
- A mathematician
- Mathematics is our one and only strategy for understanding the complexity of nature
Ralph Abraham
-Mathematics is the science of quantity
-Philosopher, and polymath
Aristotle
-The science of indirect measures
Auguste Cosme
-Mathematics is the language in which God has written in the universe
Galielo Galilei
-Mathematics is the classification and study of all possible patterns and relationships
Walter Warwick Sawyer
-Mathematics is a formal system of thought or recognizing, classifying, and exploiting patterns
Ian stewart
Learning Mathematics stimulates —– development, enchancin memory, attention, and analytical abilities.
Cognitive Development
Types of variable
Random variable
Discrete Variable
Continous Variable
Constant Variable
Parameter Variable
Independent Variable
Dependent Variable
A variable that takes on different values based on the outcomes of a random process
Random Variable
A variable that can take on a finite or countable numbers of values.
Discrete Variable
A variable that can take on any value within a given range, often involving decimals
Continuous Variable
A value that does not change within a given context or equation.
Constant Variable
A variable that remains constant within a specific context but can change when the context changes.
Parameter Variable
A variable that is manipulated or chosein in an equation or function
Independent variable
A variable whose value depends on the independent variable
Dependent Variable
Mathematics is not just about numbers; it is a way of thinking, a method of problem solving, and a tool for understanding the world.
Importance of mathematics
Are symbols or letters used to represent numbers or other mathematical objects.
Variables.
Definition of a variable:?
Purpose Of variable:?
A variable is a symbol used to reprsent a number
Variables enable generalazation in mathematics
It provides of formal way to describe collection of objects and their relationships
Language of sets.
A — is a collection of distinct objects considered as a whole.
Set
Sets are typically denoted by curly braces.
Notation
A —— is an individual object within a set.
Element
Contains no elements.
Empty set
Lists all elements of the set explicitly. For example B=(a,b,c)
Set Notation
Types of Relation
Symmetric Relation
Reflexive Relation
Transitive Relation
A relation 𝑅 R on a set 𝐴 A is called ——- if every element is related to itself. For every 𝑎 ∈ 𝐴 a∈A, ( 𝑎 , 𝑎 ) ∈ 𝑅 (a,a)∈R.
Example: The relation “is equal to” on the set of real numbers, since every number is equal to itself.
Reflexive Relation
A relation 𝑅 R on a set 𝐴 A is called —— if whenever ( 𝑎 , 𝑏 ) ∈ 𝑅 (a,b)∈R, then ( 𝑏 , 𝑎 ) ∈ 𝑅 (b,a)∈R for all 𝑎 , 𝑏 ∈ 𝐴 a,b∈A.
Example: The relation “is married to” on a set of people is symmetric because if person A is married to person B, then person B is also married to person A.
Symmetric Relation
A relation 𝑅 R on a set 𝐴 A is —– if whenever ( 𝑎 , 𝑏 ) ∈ 𝑅 (a,b)∈R and ( 𝑏 , 𝑐 ) ∈ 𝑅 (b,c)∈R, then ( 𝑎 , 𝑐 ) ∈ 𝑅 (a,c)∈R for all 𝑎 , 𝑏 , 𝑐 ∈ 𝐴 a,b,c∈A.
Example: The relation “is less than” (<) on the set of real numbers is transitive because if 𝑎 < 𝑏 a<b and 𝑏 < 𝑐 b<c, then 𝑎 < 𝑐 a<c.
Transitive Relation
This relation is a?
X= -2, -1, 0, 4, 5
Y= 0, -2, 3, -1, -3
This relation is a Function
(-2,1), (-2,3), (0,-3), (1,4), (3,1) Determine the domain/range
Domain/x= -2, 0, 1, 3
Range/y = 1, 3, -3, 4,
This relation is ?
x= -3, -1, 0, 5, 5
y= 7, 5, -2, 9, 3
This relation is a Not Function
Because there are repititions or duplicates of x values with different y values.
Is the set of all x or input values. We may describe it as the colleciton of the FIRST values in the ordered pairs
Domain
The set of all y Output values. we may describe it as the part of the collection of the second values in the ordered Pairs
Range
OBJECTS THAT WE USE IN MATH?
Numbers
Variables
Operations
Sets
Relations
Functions
Properties of Real numbers
Closure
Commutative
Associative
Distributive
Identity + x
Inverse + x
Definition: Given real numbers a and b (a, bE R), then 、1 a + b is a real number (a + b E R). Therefore, the set of reals is CLOSED with respect to addition. ab is a real number (ab E R). Therefore, tho set of reals is CLOSED with respect to multiplication.
Closure Property
Changing the order of the numbers in addition os
multiplication will not change the result.
Commutative Property
States:
a+b = b+a
Ex: 2 + 3 = 3 + 2
Commutative Property of addition
States:
a+(b+c) = (a+b)+c
3+(4+5)=(3+4)+5
Associative Property of addition
states:
ab = ba
*Ex. 4 x 5 = 5 x 4
Commutative property of multiplication
Changing the GROUPING of the numbers in addition or multiplication will not change the result.
Associative Property
States:
(ab)c= a(bc)
(2x3) x 4 = 2x (3x4)
Associative Property of multiplication
Multiplication Distributes over Addition
a(b+c) = Ab+ac
3(2+5) = 3x2 + 3x5
Distributive Property
There exists a unique number 0.
In other words adding zero to a number does not change its value
a+0 = a and 0+a =a
Additive identity property
For each real number there exist a real number such as -a their sum is zero
IN other words opposites add to zero 0
a + (-a) = 0
Additive Inverse Property
There exists a unique number 1 such that the number 1 preserves identities under multiplication
In other words multiplying a number by 1 does not change the value of the number
a x 1 = a and 1 x a = a
Multiplicative Identity Property
For each number there exist a unique real number 1/a such that ther product is 1.
a x 1/a = 1
Multiplicative Inverse Property
