Set Notation and Probability Basics

Set Language and Probability

• 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

Functions

• 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

Circle Theorems

• 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

Sequence

• 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)

Trigonometry

• 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)

Transformations

• 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

Statistics

• 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

Mensuration

• 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

Set Language and Probability

• 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

Functions

• 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

Circle Theorems

• The angle at the centre of a circle is twice the angle at the circumference
• Angles in the same segment of a circle are equal
• The opposite angles of a cyclic quadrilateral add up to 180°
• The tangent to a circle is perpendicular to the radius at the point of contact

Sequences

• An arithmetic sequence has a common difference between consecutive terms
• A geometric sequence has a common ratio between consecutive terms
• The formula for an arithmetic sequence is an = a1 + (n-1)d
• The formula for a geometric sequence is an = ar^(n-1)

Trigonometry

• Sine (sin) is the ratio of the opposite side to the hypotenuse
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse
• Tangent (tan) is the ratio of the opposite side to the adjacent side
• Pythagorean identities: sin^2(A) + cos^2(A) = 1 and tan^2(A) + 1 = sec^2(A)

Transformations

• 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

Statistics

• 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

Mensuration

• The perimeter of a shape is the distance around it
• The area of a shape is the amount of space inside it
• The volume of a 3D shape is the amount of space inside it
• The formula for the perimeter of a rectangle is 2(l + w)
• The formula for the area of a rectangle is lw
• The formula for the volume of a cube is s^3