What is Set Theory?

Mathematics is full of ways to organise and group things — and set theory is one of the most fundamental. It gives us a precise language for talking about collections of objects, whether those objects are numbers, shapes, living organisms, or anything else we can define clearly.

Set theory was developed in the late 19th century by mathematician Georg Cantor, and it now forms a foundation for almost all of modern mathematics. In MYP Sciences and Mathematics, we use set theory to model real-world situations — for example, grouping chemicals by their properties, or classifying organisms by shared characteristics.

Set

A set is a well-defined collection of distinct objects. The objects in a set are called elements or members.

The word well-defined is crucial: it means there is no ambiguity about whether something belongs to the set or not. For example, "all even numbers between 1 and 10" is well-defined, but "all tall students" is not — because "tall" is subjective.

Analogy

Think of a set like a labelled bag. You decide exactly what goes in the bag based on a clear rule. Anything that fits the rule goes in; anything that doesn't, stays out. The bag itself is the set, and the items inside are the elements.

Sets as scientific models:

In the MYP Sciences unit Thinking With Models, we study how models help us understand the world. A set is a type of model — it simplifies reality by grouping objects according to chosen rules. Like all models, sets have assumptions and limitations:

The membership rule is chosen by us — a different scientist might draw the boundaries differently.
Real-world objects are complex; placing them in a set ignores properties not relevant to our rule.
Some objects are genuinely hard to classify (e.g., viruses: are they living or non-living?).

Recognising these limitations is part of thinking scientifically.

Note

MYP Key Concepts connection: Set theory directly supports the key concept of Systems (understanding how parts relate to a whole) and the related concept of Models (using simplified representations to describe reality). When you define a set, you are building a model of a category in the real world.

Sets in Scientific Modelling

Before diving into notation, it helps to see why set theory matters in science. Whenever we classify, sort, or group things in the natural world, we are using set theory — whether we realise it or not.

Applications in science:

Biology: Classifying organisms into kingdoms, phyla, classes — each is a set with membership rules. The boundaries can be fuzzy (e.g., classifying viruses is genuinely debated).
Chemistry: Grouping elements by properties (metals, non-metals, noble gases) — these are sets, and some properties overlap (e.g., metalloids share properties of both metals and non-metals).
Physics: Categorising waves as longitudinal or transverse; categorising forces as contact or non-contact.
Environmental science: Identifying which species live in overlapping habitats — a classic Venn diagram application.

Example

Modelling with sets — Periodic Table:

Let $U$ = first 20 elements of the periodic table

Let $M$ = metals in $U$ = $Li, Be, Na, Mg, Al, K, Ca$

Let $S$ = solid elements at room temperature within $U$ = $Li, Be, B, C, Na, Mg, Al, Si, P, S, K, Ca$

(Note: Cl is a gas at room temperature so it is excluded from $S$ . H, He, N, O, F, Ne, Cl, Ar are gases or liquids at room temperature.)

Then $M \cap S$ = metals that are solid at room temperature — most of the metals in the first 20 elements!

And $M^{'}$ = non-metals and metalloids in the first 20 elements.

This kind of thinking helps scientists organise vast amounts of data clearly and precisely.

Models have limitations — so do sets:

When we put chlorine into a set called "non-metals," we are making a modelling choice. Chlorine at room temperature is a gas, but at very low temperatures it becomes a solid. Our set $S$ above assumes standard room temperature — that assumption is part of the model. Change the assumption, and the set membership changes.

Note

In MYP Sciences, when you design an investigation, you are essentially defining your sample set, your control set, and your variable conditions — all set-theory concepts in disguise. Criterion B (Inquiring and designing) involves deciding exactly which objects, organisms, or trials belong in each group — that is set-theory thinking applied to real scientific inquiry.

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