Netherlands VWO 5 Mathematics C
Chapters: 3
1. Relationships
Relationships and their representations
- Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
2. Logical reasoning
Logical reasoning
- Logical Reasoning – Logical reasoning means getting from facts (premises) to a conclusion in a way we can check. A statement is either true or false. Statements are joined with NOT, AND, OR, IF…THEN and IF AND ONLY IF, and truth tables show when the result is true. In deduction, if the premises are true and the form is valid, the conclusion must be true (modus ponens, modus tollens, syllogisms). Venn diagrams test syllogisms with 'all', 'no' and 'some'. Induction and analogy go from examples to a general idea: the conclusion is only probable. A direct proof goes step by step from what is known; an indirect proof assumes the opposite and reaches a contradiction. Fallacies are tempting but faulty arguments.
3. Statistics and probability (part 2)
Probability · Probability distributions
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.