Math Basics: Addition, Subtraction, Multiplication, Division

Definition: Combining two or more numbers to get a total sum.
Notation: ( a + b ) (where ( a ) and ( b ) are numbers).
Properties:
- Commutative: ( a + b = b + a )
- Associative: ( (a + b) + c = a + (b + c) )
- Identity: ( a + 0 = a )

Definition: Finding the difference between two numbers.
Notation: ( a - b ) (where ( a ) is the minuend and ( b ) is the subtrahend).
Properties:
- Not commutative: ( a - b \neq b - a )
- Not associative: ( (a - b) - c \neq a - (b - c) )
- Identity: ( a - 0 = a )

Definition: Repeated addition of a number.
Notation: ( a \times b ) or ( ab ).
Properties:
- Commutative: ( a \times b = b \times a )
- Associative: ( (a \times b) \times c = a \times (b \times c) )
- Identity: ( a \times 1 = a )
- Zero Property: ( a \times 0 = 0 )

Definition: Splitting a number into equal parts or finding how many times one number contains another.
Notation: ( a \div b ) or ( \frac{a}{b} ) (where ( a ) is the dividend and ( b ) is the divisor).
Properties:
- Not commutative: ( a \div b \neq b \div a )
- Not associative: ( (a \div b) \div c \neq a \div (b \div c) )
- Identity: ( a \div 1 = a )
- Division by zero is undefined.

Definition: Combining two functions where the output of one function becomes the input of another.
Notation: If ( f ) and ( g ) are functions, the composition is written as ( (f \circ g)(x) = f(g(x)) ).
Properties:
- Not commutative: ( f \circ g \neq g \circ f )
- Associative: ( f \circ (g \circ h) = (f \circ g) \circ h )
- Identity: ( f \circ I = f ) and ( I \circ f = f ) (where ( I ) is the identity function).
Example:
- If ( f(x) = 2x ) and ( g(x) = x + 3 ), then ( (f \circ g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6 ).

Combining two or more numbers results in a total sum.
Notation is expressed as ( a + b ) (with ( a ) and ( b ) as numbers).
Commutative property: The order of addition does not affect the result (( a + b = b + a )).
Associative property: Grouping of numbers does not impact the result (( (a + b) + c = a + (b + c) )).
Identity property: Adding zero to any number does not change its value (( a + 0 = a )).

Finding the difference between two numbers defines subtraction.
Notation is presented as ( a - b ) (where ( a ) is the minuend and ( b ) is the subtrahend).
The operation is not commutative: The order matters (( a - b \neq b - a )).
Subtraction is not associative: Changing grouping affects the result (( (a - b) - c \neq a - (b - c) )).
Identity property: Subtracting zero from any number leaves it unchanged (( a - 0 = a )).

Defined as repeated addition of a number.
Can be denoted as ( a \times b ) or ( ab ).
Multiplication is commutative: Order does not change the product (( a \times b = b \times a )).
Associative property: Grouping does not affect the product (( (a \times b) \times c = a \times (b \times c) )).
Identity property: Multiplying any number by one does not change its value (( a \times 1 = a )).
Zero property: Any number multiplied by zero results in zero (( a \times 0 = 0 )).

Operation of dividing a number into equal parts or determining how many times one number fits into another.
Notation is written as ( a \div b ) or ( \frac{a}{b} ) (with ( a ) as the dividend and ( b ) as the divisor).
Division is not commutative: The order matters (( a \div b \neq b \div a )).
Not associative: Changing grouping alters the result (( (a \div b) \div c \neq a \div (b \div c) )).
Identity property: Dividing a number by one keeps its value unchanged (( a \div 1 = a )).
Division by zero is undefined and cannot be performed.

Combining functions such that the output of one serves as the input for another defines function composition.
Notation includes ( (f \circ g)(x) = f(g(x)) ).
Composition is not commutative: The order of functions affects the outcome (( f \circ g \neq g \circ f )).
Associative property holds: Grouping of functions does not change the composite outcome (( f \circ (g \circ h) = (f \circ g) \circ h )).
Identity property: Composing a function with the identity function returns the original function (( f \circ I = f ) and ( I \circ f = f )).
Example: For ( f(x) = 2x ) and ( g(x) = x + 3 ), composition gives ( (f \circ g)(x) = 2(x + 3) = 2x + 6 ).