Set Notation and Probability Basics

Set Notation:
- Union (A ∪ B): all elements in A or B or both
- Intersection (A ∩ B): all elements in both A and B
- Complement (A'): all elements not in A
- Difference (A - B): all elements in A but not in B
Set Operations:
- Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Probability:
- Experiment: an action or situation that can produce a set of outcomes
- Sample Space: the set of all possible outcomes
- Event: a subset of the sample space
- Probability of an event: P(A) = number of favorable outcomes / total number of outcomes

Domain: the set of all possible input values
Range: the set of all possible output values
Function Notation: f(x) = output value for input x
Types of Functions:
- One-to-one: each output value corresponds to exactly one input value
- Many-to-one: each output value corresponds to more than one input value
- Onto: every output value is reached by at least one input value

Angle at the Centre: the angle at the centre of a circle is twice the angle at the circumference
Angles in the Same Segment: angles in the same segment of a circle are equal
Cyclic Quadrilateral: the opposite angles of a cyclic quadrilateral add up to 180°
Tangents and Radii: the tangent to a circle is perpendicular to the radius at the point of contact

Arithmetic Sequence: a sequence with a common difference between consecutive terms
Geometric Sequence: a sequence with a common ratio between consecutive terms
Formulae:
- Arithmetic sequence: an = a1 + (n-1)d
- Geometric sequence: an = ar^(n-1)

Trigonometric Ratios:
- Sine (sin): opposite side / hypotenuse
- Cosine (cos): adjacent side / hypotenuse
- Tangent (tan): opposite side / adjacent side
Pythagorean Identities:
- sin^2(A) + cos^2(A) = 1
- tan^2(A) + 1 = sec^2(A)

Translations: a shift of a graph by a certain number of units in the x and y directions
Rotations: a rotation of a graph by a certain number of degrees around the origin
Reflections: a reflection of a graph over the x-axis, y-axis, or a certain line
Enlargements: a scale change of a graph by a certain factor

Measures of Central Tendency:
- Mean: the average value of a dataset
- Median: the middle value of a dataset when arranged in order
- Mode: the most frequently occurring value in a dataset
Measures of Dispersion:
- Range: the difference between the largest and smallest values
- Interquartile Range (IQR): the difference between the upper and lower quartiles

Perimeter: the distance around a shape
Area: the amount of space inside a shape
Volume: the amount of space inside a 3D shape
Formulae:
- Perimeter of a rectangle: 2(l + w)
- Area of a rectangle: lw
- Volume of a cube: s^3

A union of two sets A and B (A ∪ B) includes all elements in A or B or both
The intersection of two sets A and B (A ∩ B) includes all elements in both A and B
The complement of a set A (A') includes all elements not in A
The difference of two sets A and B (A - B) includes all elements in A but not in B
Set operations are commutative and associative
Distributive property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Probability of an event is the number of favorable outcomes divided by the total number of outcomes
An experiment is an action or situation that can produce a set of outcomes
A sample space is the set of all possible outcomes
An event is a subset of the sample space

The domain of a function is the set of all possible input values
The range of a function is the set of all possible output values
Function notation: f(x) represents the output value for input x
One-to-one functions have each output value corresponding to exactly one input value
Many-to-one functions have each output value corresponding to more than one input value
Onto functions have every output value reached by at least one input value

Translations involve shifting a graph by a certain number of units in the x and y directions
Rotations involve rotating a graph by a certain number of degrees around the origin
Reflections involve reflecting a graph over the x-axis, y-axis, or a certain line
Enlargements involve scaling a graph by a certain factor

The mean is the average value of a dataset
The median is the middle value of a dataset when arranged in order
The mode is the most frequently occurring value in a dataset
The range is the difference between the largest and smallest values
The interquartile range (IQR) is the difference between the upper and lower quartiles