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Sets are collections of distinct objects where membership (∈) denotes an element belongs and non-membership (∉) denotes it does not. Sets can be described in roster form {a,b,c}, set-builder notation {x | P(x)}, or by properties. The order and repetition of elements in notation are irrelevant to the set itself.
Start with concrete examples: {1,2,3}, {vowels}, ∅. Practice translating between roster and set-builder notation: {2,4,6,8} = {n | n is even and 1 ≤ n ≤ 8}. Verify membership: 5 ∉ {primes ≤ 3}.
A set is simply a collection of distinct objects — called its elements or members — treated as a single mathematical entity. The symbol ∈ encodes membership: writing 3 ∈ {1, 2, 3} asserts that 3 is an element of the set, while 5 ∉ {1, 2, 3} asserts it is not. The empty set ∅ = {} is the unique set with no elements at all; it is a legitimate set, not nothing.
Two features distinguish sets from lists or sequences. First, repetition is ignored: {1, 1, 2} and {1, 2} are the same set because a set either contains an element or it does not — there is no notion of multiplicity. Second, order is ignored: {1, 2, 3} and {3, 1, 2} are the same set. These two rules follow from the axiom of extensionality: two sets are equal if and only if they have exactly the same members, regardless of how they are written.
Sets can be described in two main ways. Roster form lists the elements explicitly: {2, 4, 6, 8}. Set-builder notation describes them by a property: {x | x is a positive even integer less than 10}. The vertical bar reads as "such that." Set-builder notation is indispensable when a set is too large to list (e.g., {x | x is a prime number}) or is defined by a condition rather than an enumeration.
A common source of confusion is the overlap in notation between sets and other mathematical objects. The curly braces {a, b} denote a set; the round parentheses (a, b) denote an ordered pair or an open interval. These are entirely different structures. In an ordered pair, (3, 5) ≠ (5, 3). In a set, {3, 5} = {5, 3}. Keeping this distinction clear is essential as you encounter Cartesian products and relations, where both notations appear together.
Set membership is the simplest and most primitive relation in mathematics. Every structure you will encounter in formal set theory — subsets, functions, ordered pairs, numbers — is ultimately built from sets and the ∈ relation. Mastering the notation now pays compounding dividends: when you later define the natural numbers as sets, or describe a function as a set of ordered pairs, the logic will feel natural rather than arbitrary.