Year 12: Discrete mathematics

Discrete mathematics focuses on systems that change step by step rather than continuously. Recursive sequences and matrices both reward careful notation and a clear view of what each term or entry means.

Always attach meaning to the object you are using. A matrix entry, a row and a recursive rule should all be interpretable in context, not just manipulated symbolically.

What to master

Reading and building recursive rules for sequences.
Finding matrix elements from position rules or given transitions.
Using transition and Leslie matrices to model repeated changes over time.
Interpreting permutation, communication and dominance matrices in context.

Exam habit: when a question gives a real-world matrix model, write one sentence about what rows and columns represent before multiplying anything.

Discrete Mathematics tutorials

How to read discrete models

Discrete mathematics is about step-by-step change. That means every object in the question needs an interpretation. A recursive formula tells you how one term is built from the previous term. A transition matrix tells you how a system moves from one stage to the next. A permutation matrix rearranges entries without changing their values. If you keep asking what the mathematical object means, the topic becomes far easier than trying to memorise matrix types as isolated facts.

Worked example

Problem: A sequence is defined by

u_{1} = 4

and

u_{n + 1} = u_{n} + 3

. Find the first four terms.

The first term is given: $u_{1} = 4$ .
Use the rule once: $u_{2} = 4 + 3 = 7$ .
Use the rule again: $u_{3} = 7 + 3 = 10$ .
Once more: $u_{4} = 10 + 3 = 13$ .

Answer: the first four terms are

4, 7, 10, 13

Common traps

Using a matrix mechanically without saying what the entries represent.
Applying a recursive rule to the wrong term number.
Multiplying matrices in the wrong order when a model is directional.

Study routine

The best revision method for this topic is to read every symbolic statement aloud in words. If you can explain what a transition or recursive step means without looking at notes, you usually understand the mathematics behind it rather than just the notation.

That is especially important with matrices, because rows, columns, and entries can all represent different things depending on the model. If you know what each position means, matrix multiplication becomes interpretation as much as arithmetic. If you do not know what the entries mean, even correct multiplication may answer the wrong question.

Revision focus

These topics reward careful structure more than fast calculation. In complex numbers, discrete mathematics, graph theory, and number theory, the key objects carry meaning that should stay visible all the way through the working. Real and imaginary parts, matrix entries, graph properties, and modular statements all behave best when they are written clearly and interpreted consistently.

A reliable way to revise is to explain each line of working in words as you go. If you can say what a power of i is doing, what a matrix row represents, what a graph property implies, or what a congruence statement means, then the notation is working for you rather than against you. That level of clarity usually prevents the small symbolic errors that turn easy marks into lost marks.