A set is a well-defined collection of distinct objects, called elements or members. Sets are written with curly braces: {1, 2, 3} is the set containing 1, 2, and 3. The symbol ∈ means "is an element of" (3 ∈ {1, 2, 3}) and ∉ means "is not an element of" (5 ∉ {1, 2, 3}). Order and repetition do not matter: {3, 1, 2} = {1, 2, 3} and {1, 1, 2} = {1, 2}. Sets can be described by listing elements (roster notation) or by stating a rule (set-builder notation): {x | x is an even number less than 10} = {2, 4, 6, 8}. The empty set ∅ = {} contains no elements.
Start with collections students already understand: the set of planets, the set of vowels, the set of prime numbers less than 20. Practice writing sets in roster form and translating to set-builder notation and back. Emphasize the two defining rules of sets: order does not matter, and duplicates are ignored. Introduce the empty set with examples: "the set of even prime numbers greater than 2" is ∅.
You have sorted and classified objects before — grouping animals by type, numbers by property, shapes by sides. A set is the mathematical formalization of a group. It is a collection of distinct objects treated as a single entity, and it comes with precise notation for describing what is in the collection and what is not.
The most basic notation is roster form: you list the elements inside curly braces, separated by commas. {2, 3, 5, 7} is the set of single-digit prime numbers. The curly braces signal "this is a set," and the elements are the specific objects in it. Two fundamental rules govern sets. First, order is irrelevant: {2, 3, 5, 7} = {7, 5, 3, 2} = {3, 7, 2, 5}. Second, repetition is irrelevant: {2, 2, 3} = {2, 3}. A set either contains an object or it does not — there is no "contains it twice."
The membership symbol ∈ states that an element belongs to a set: 3 ∈ {1, 2, 3} says "3 is an element of the set {1, 2, 3}." The negation ∉ says the opposite: 4 ∉ {1, 2, 3}. These symbols give you a precise language for making claims about membership, which will become essential as you learn about subsets, intersections, and other set operations.
When a set is too large to list or is defined by a pattern, you use set-builder notation: {x | condition on x}. The vertical bar reads "such that." So {x | x is a prime number less than 20} describes the set {2, 3, 5, 7, 11, 13, 17, 19} without listing each element. You can also use set-builder notation for infinite sets: {x | x is a positive even integer} = {2, 4, 6, 8, ...}, which no roster could fully write out.
The empty set, written ∅ or {}, is the set with no elements. It is not the same as "nothing" — it is a real set; it is just empty. Think of it like an empty box: the box exists, you can talk about it, you can compare it to other boxes — it just has nothing inside. The empty set comes up naturally: "the set of even prime numbers greater than 2" is empty because 2 is the only even prime. Every set has the empty set as a subset, which is a concept you will encounter soon.