Making sense
of sets.

Think of a box containing red pencils. A set is the mathematical version of a collection: we need a clear rule for deciding which objects belong. Each object in the collection is called an element.

The rule must be definite. “Students older than 14” can be checked; “interesting students” cannot be decided in the same precise way. We usually give a set a capital letter and list its elements inside curly brackets. The brackets describe membership, not an addition calculation.

Set symbols are abbreviations for ordinary sentences. Before learning their shapes, say what you want to describe: an object belongs, two collections overlap, or one collection sits entirely inside another.

An element and a set play different roles. The number 3 is an element; {3} is a set containing one element. Keep that distinction in mind as you read the notation table. Translate each symbolic statement into words before deciding whether it is true.

Imagine sorting students by whether they play football and whether they swim. Some do both, so two separate lists would hide an important overlap. A Venn diagram gives each possible membership a region.

Place someone who does both activities in the overlap, not once in each outer region. The union includes everyone in either circle; the intersection includes only the shared part. When adding circle totals, notice that an overlap member appears in both totals even though it is only one person.

With three activities, a student can belong to none, one, two or all three. The central region belongs to every circle, so it affects each pairwise overlap as well as each individual total.

Fill the smallest, most specific region first: all three sets. Next find the pair-only regions by subtracting the centre from the stated pair totals. Only then calculate the single-set regions. This order prevents a student being counted twice within one circle.

A long list is not always the easiest way to describe a set. Instead, we can give a condition that every member must satisfy. This is called set-builder notation, and it should be read as a sentence.

The chosen number domain matters. The integers between 2 and 5 form a short list, but the real numbers between them also include values such as 2.1 and 3.75. When comparing sets, a subset statement asks whether every element of the first is also in the second.

Use a small sketch while practising, even if the question only contains symbols. Write the universal set first, because “not in A” has no fixed meaning until we know the collection we are choosing from.

Read each word carefully: “only”, “both”, “exactly” and “at least” select different regions. After filling a counting diagram, add every separate region, including the outside, and check that the result equals the total population.