𧩠Sets & Matrices
Sets and Matrices cover set notation, Venn diagrams, union/intersection, matrix operations, and inverse matrices. These topics are relatively newer additions to the E-Math syllabus and are very scorable with consistent practice. Many students find sets intuitive, and matrix questions follow predictable patterns.
Sets questions often appear as word problems involving surveys, club memberships, or student activities. The key tool is the Venn diagram, which provides a visual way to organise information about overlapping groups. Matrix questions typically test arithmetic operations and the ability to interpret matrices in real-world contexts (e.g., quantities and prices).
Set Notation & Terminology
A set is a well-defined collection of distinct objects called elements or members. We use curly brackets to list elements: A = {1, 2, 3, 4, 5}. The symbol means "is an element of" and means "is not an element of". The universal set (denoted by a script E or U) contains all elements under consideration in a particular problem.
Understanding set notation precisely is crucial because exam questions often ask you to shade regions on Venn diagrams or describe sets using notation. A common exam task is to express a shaded region using union, intersection, and complement symbols.
Set Notation Essentials
Union (A βͺ B)
Elements in A OR B (or both). On a Venn diagram, this is everything inside either circle.
Intersection (A β© B)
Elements in BOTH A AND B simultaneously. The overlapping region in a Venn diagram.
Complement (A')
Elements NOT in A but still in the universal set. Everything outside circle A.
Subset (A β B)
Every element of A is also in B. Circle A is entirely inside circle B. Also written A β B.
Empty Set (β )
A set with no elements. A β© B = β means A and B have no overlap (mutually exclusive / disjoint).
n(A)
The number of elements in set A. For example, if A = {2, 4, 6}, then n(A) = 3.
Key Venn Diagram Formula
For two sets A and B: . We subtract the intersection because those elements are counted twice (once in A and once in B). This formula is the basis for solving almost all Venn diagram word problems.
πWorked Example 1: Two-Set Venn Diagram
In a class of 40 students, 25 play football, 18 play basketball, and 8 play both. How many play (a) at least one sport? (b) neither sport? (c) football only?
Venn diagram showing union, intersection, and complement regions
πWorked Example 2: Three-Set Venn Diagram
In a survey of 100 people: 60 like tea, 50 like coffee, 30 like juice, 20 like both tea and coffee, 15 like both tea and juice, 10 like both coffee and juice, 5 like all three. Find how many like none of them.
Three-set Venn diagram showing all intersection regions
πWorked Example 3: Shading Venn Diagrams
Shade the region representing on a three-set Venn diagram.
Three-set Venn diagram showing all intersection regions
If n(A) = 12, n(B) = 8, and n(A intersection B) = 3, what is n(A union B)?
Matrix Operations
A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is written as rows x columns. For example, a 2 x 3 matrix has 2 rows and 3 columns. For O-Level, you need to know: addition, subtraction, scalar multiplication, matrix multiplication, and inverse of a 2 x 2 matrix.
Addition/Subtraction: Only possible when matrices have the same order. Add (or subtract) corresponding elements. Scalar Multiplication: Multiply every element by the scalar. Matrix Multiplication: The number of columns of the first matrix must equal the number of rows of the second. The result has the same number of rows as the first matrix and the same number of columns as the second.
Matrix Multiplication Rule
You can only multiply an matrix by an matrix. The result is an matrix. The inner dimensions must match. Matrix multiplication is NOT commutative: AB is not always equal to BA.
πWorked Example 4: Matrix Multiplication
Calculate the product of the 2x2 matrix [2 3; 1 4] and the 2x1 matrix [5; 2].
πWorked Example 5: 2x2 Matrix Multiplication
Calculate [1 2; 3 4] x [5 6; 7 8].
Inverse of a 2 x 2 Matrix
The identity matrix I is the matrix equivalent of the number 1 in multiplication. For 2 x 2 matrices, I = [1 0; 0 1]. The inverse of matrix A, written , satisfies . Not every matrix has an inverse β a matrix is singular (no inverse) when its determinant is zero.
Inverse of a 2x2 Matrix
Determinant
πWorked Example 6: Finding the Inverse Matrix
Find the inverse of A = [3 1; 5 2].
πWorked Example 7: Using Inverse Matrix to Solve Equations
Solve using matrices: and .
A 3x2 matrix is multiplied by a 2x4 matrix. What are the dimensions of the result?
β οΈCommon Mistakes β Sets & Matrices
Why: Students mix up interior and exterior angle formulas. Interior + Exterior = 180 degrees for each vertex.
Why: AAA proves similarity only, not congruence. Congruence requires SSS, SAS, AAS, or RHS.
Matrices in Real-World Context
Exam questions often present matrices in a practical context β such as prices and quantities, routes between locations, or scores in a competition. You need to interpret what the matrix represents, perform the required operation, and then interpret the result in context.
πWorked Example 8: Matrices in Context
A shop sells 3 types of drink. Matrix Q shows quantities sold on Mon and Tue. Matrix P shows the unit price ($). Find QP and interpret it.
πWorked Example 9: Finding a Missing Element
Given [2 k; 3 5] has determinant 1. Find k.
If matrix A = [2 1; 4 3], what is det(A)?
πWorked Example 10: Solving Simultaneous Equations Using Matrices
Using matrices, solve the simultaneous equations: and .
Set Builder Notation and Number Sets
Sets can also be described using set builder notation, which defines a set by a property that its members share. For example, A = {x : x is a prime number less than 20} describes all primes less than 20. You should also know the standard number sets: natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R).
πWorked Example 11: Set Builder Notation
List the elements of A = {x : x is an integer and -2 <= x < 5}.
πWorked Example 12: Complement and De Morgan's Laws
Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {2, 4, 6, 8, 10}. B = {1, 2, 3, 4, 5}. Find (a) (A union B)' (b) A' intersection B'
Venn diagram showing union, intersection, and complement regions
De Morgan's Laws
Two important identities for set complements:
- (A union B)' = A' intersection B' β the complement of the union equals the intersection of complements
- (A intersection B)' = A' union B' β the complement of the intersection equals the union of complements
These laws are useful for simplifying complex set expressions and are sometimes tested directly.
U = {1,2,3,...,10}. A = {1,3,5,7,9}. B = {3,6,9}. Find A intersection B.
Sets & Matrices Are Scoring Topics!
These topics follow very predictable patterns. Venn diagram word problems always use the same formula: n(A union B) = n(A) + n(B) - n(A intersection B). Matrix multiplication always follows the same "row times column" process. Practise 5-10 questions and you will handle any exam variation confidently.