What you'll learn
Sets is one of the most reliably scoring topics in CXC CSEC Mathematics, because the notation is finite, the diagrams are straightforward, and the problem types repeat from year to year. A set is simply a collection of well-defined objects, and almost every question reduces to organising information into a Venn diagram and then reading values off it. The topic also appears in Paper 2 as a full question carrying substantial marks, often combined with a practical word problem about students studying subjects or people using services. By the end of this guide you should be able to use all the set notation correctly, list and describe sets in both forms, find unions, intersections and complements, draw and interpret Venn diagrams for two and three sets, solve problems involving unknown quantities in the overlaps, and use the formula relating the sizes of two sets and their union.
Key terms and definitions
Set — a collection of well-defined objects, written inside curly brackets
Element — a member of a set; the symbol for "is an element of" is ∈ and for "is not an element of" is ∉
Universal set — the set containing all elements under consideration, denoted U
Empty set — a set with no elements, written { } or ∅
Finite set — a set with a countable number of elements
Infinite set — a set whose elements cannot be counted
Subset — a set whose every element belongs to another set; the symbol is ⊂
Union — all elements in either set or in both, symbol ∪
Intersection — only the elements in both sets, symbol ∩
Complement — all elements in the universal set that are not in the given set, written A′
Disjoint sets — sets with no elements in common, so their intersection is empty
Cardinality — the number of elements in a set, written n(A)
Core concepts
Describing sets
A set can be described in two ways, and questions may ask for either.
The roster or listing method writes out the elements: A = {2, 4, 6, 8, 10}.
The set-builder or descriptive method states the rule: A = {even numbers between 1 and 11}.
Elements are never repeated, and the order in which they are written does not matter. The set {1, 2, 3} is identical to the set {3, 2, 1}.
The cardinality of a set is the number of elements it contains. For the set A above, n(A) = 5.
The basic operations
The union of A and B, written A ∪ B, contains every element that is in A, or in B, or in both. Think of it as combining the two sets, listing each element only once.
The intersection of A and B, written A ∩ B, contains only those elements found in both sets at the same time.
The complement of A, written A′, contains every element of the universal set that is not in A.
A useful memory aid: union is associated with the word "or" and takes everything, while intersection is associated with "and" and takes only the overlap. The symbol ∪ resembles the first letter of union, which helps when the two are confused under pressure.
Two sets with nothing in common are disjoint, and their intersection is the empty set.
Subsets
A is a subset of B if every element of A is also an element of B. Every set is a subset of itself, and the empty set is a subset of every set.
The number of subsets of a set containing n elements is 2 raised to the power n. A set with 3 elements therefore has 8 subsets, and this result appears in multiple choice questions regularly.
Venn diagrams for two sets
A Venn diagram represents the universal set as a rectangle, with each set drawn as a circle inside it.
Two overlapping circles divide the rectangle into four distinct regions, and identifying them is the key to every problem of this type.
The overlap contains elements in both A and B.
The part of circle A outside the overlap contains elements in A only.
The part of circle B outside the overlap contains elements in B only.
The area inside the rectangle but outside both circles contains elements in neither set.
The single most important technique in this topic is to fill in the overlap first and work outwards. Doing so prevents the commonest error, which is double-counting the elements that belong to both sets.
Solving two-set problems
A typical question states that a number of students study Mathematics, a number study Physics, a number study both, and asks how many study neither.
The method is systematic. Place the number who study both in the overlap. Subtract that number from the Mathematics total to find those studying Mathematics only, and place it in the left region. Subtract it from the Physics total to find those studying Physics only, and place it in the right region. Add the three regions and subtract from the total number of students to find how many study neither.
The relationship can also be written as a formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The subtraction is necessary because the elements in the intersection have been counted once in n(A) and once in n(B), so one of those counts must be removed.
This formula is worth memorising, since some questions supply n(A ∪ B) and ask for the intersection, which requires rearranging rather than drawing.
Problems with an unknown in the overlap
A harder and very common variant gives the total and both set sizes but not the number in the intersection, which must be found.
Let the number in the overlap be x. Then the number in A only is n(A) − x, and the number in B only is n(B) − x. Adding all regions, including those in neither set, gives an equation in x which can be solved.
Working in terms of x from the start, and labelling every region on the diagram before forming the equation, makes these questions routine. Attempting them without a diagram is where most marks are lost.
Venn diagrams for three sets
Three overlapping circles divide the rectangle into eight regions: the central region belonging to all three sets, three regions belonging to exactly two sets, three regions belonging to exactly one set, and the region outside all three.
The method is the same but the order is critical: always begin with the centre, where all three sets overlap, then fill the regions belonging to exactly two sets, then those belonging to one set only, and finally the region outside.
When filling a region for exactly two sets, remember to subtract the central value. If 12 students study both Mathematics and Physics, and 5 of those also study Chemistry, then the region for Mathematics and Physics but not Chemistry contains 12 − 5 = 7.
This subtraction is the single most frequent source of error in three-set problems, and working from the centre outwards is what prevents it.
Reading the notation in questions
Examination questions often express conditions in notation rather than words, so translating fluently matters.
n(A ∩ B) asks how many elements are in both sets. n(A ∪ B) asks how many are in at least one. n(A′) asks how many are not in A. n(A ∩ B)′ asks how many are not in both. A ∩ B′ means the elements in A but not in B, which is the "A only" region.
Reading these carefully is worth doing twice, since a misread symbol produces a fully worked but wrong answer.
Worked examples
Example 1: A two-set problem (4 marks)
In a class of 40 students, 25 study Mathematics, 18 study Physics and 10 study both. How many study neither subject?
Place 10 in the overlap, since these students study both.
Those studying Mathematics only are 25 − 10 = 15, and those studying Physics only are 18 − 10 = 8.
The number studying at least one subject is 15 + 10 + 8 = 33.
The number studying neither is 40 − 33 = 7.
As a check, applying the formula gives n(A ∪ B) = 25 + 18 − 10 = 33, and 40 − 33 = 7, which agrees.
Example 2: Finding an unknown intersection (5 marks)
In a group of 50 people, 30 drink tea, 27 drink coffee and 6 drink neither. How many drink both?
Let the number who drink both be x.
Those who drink tea only are 30 − x, and those who drink coffee only are 27 − x.
The four regions must sum to the total of 50, so (30 − x) + x + (27 − x) + 6 = 50.
Simplifying the left side gives 63 − x = 50, so x = 13.
Therefore 13 people drink both. Checking: tea only is 17, both is 13, coffee only is 14, neither is 6, and 17 + 13 + 14 + 6 = 50 as required.
Example 3: A three-set problem (5 marks)
In a survey of 60 students, 30 play cricket, 25 play football and 22 play netball. Of these, 12 play cricket and football, 9 play football and netball, 11 play cricket and netball, and 5 play all three. How many play none of the three sports?
Begin at the centre with 5, the number playing all three.
For exactly two sports, subtract the centre from each pair value. Cricket and football only is 12 − 5 = 7. Football and netball only is 9 − 5 = 4. Cricket and netball only is 11 − 5 = 6.
For exactly one sport, subtract all overlapping regions from each total. Cricket only is 30 − 7 − 6 − 5 = 12. Football only is 25 − 7 − 4 − 5 = 9. Netball only is 22 − 4 − 6 − 5 = 7.
Adding every region inside the circles gives 12 + 9 + 7 + 7 + 4 + 6 + 5 = 50.
The number playing none is 60 − 50 = 10.
Common mistakes and how to avoid them
The most frequent error is double-counting the intersection. If 25 study Mathematics and 10 study both, then only 15 study Mathematics alone, and writing 25 in the Mathematics-only region is wrong.
Students often confuse union with intersection, particularly under time pressure. Union takes everything and is associated with "or"; intersection takes only the overlap and is associated with "and".
In three-set problems, many candidates forget to subtract the central value when filling the regions for exactly two sets. Always work from the centre outwards.
Another common slip is forgetting the region outside all the circles. The elements in neither set are part of the universal set and must be included when totalling.
Finally, candidates sometimes attempt these problems without drawing a diagram. The diagram carries method marks in Paper 2 and makes errors visible, so draw it even when the arithmetic seems easy.
Exam technique for "Sets"
Draw the Venn diagram before doing any calculation, and label the universal set with the total. The diagram is worth marks in its own right and organises the information for you.
Fill the most restricted region first: the triple overlap in a three-set problem, or the double overlap in a two-set problem. Every other region is then found by subtraction.
Where an unknown is involved, let it be x and express every region in terms of x before forming the equation. Trying to reason without algebra makes these questions much harder than they are.
Always check by adding every region and confirming the total matches the universal set. This check takes seconds and catches most arithmetic errors.
Translate any set notation in the question into words before starting, and write the translation down so you do not misread it later.
Quick revision summary
A set is a collection of well-defined objects listed in curly brackets, with no repeats and no significance to order, and n(A) gives the number of elements. Union, written A ∪ B, takes all elements in either set; intersection, written A ∩ B, takes only those in both; and the complement A′ takes everything in the universal set not in A. Disjoint sets have an empty intersection, and a set of n elements has 2 to the power n subsets. A two-set Venn diagram has four regions — A only, both, B only, and neither — and the relationship n(A ∪ B) = n(A) + n(B) − n(A ∩ B) removes the double count of the overlap. Where the intersection is unknown, let it be x, express each region in terms of x, and solve the equation formed by summing all regions to the total. A three-set diagram has eight regions and must be filled from the centre outwards, subtracting the central value when completing each pair region. Always draw the diagram, include the region outside all circles, and check that every region sums to the universal set.