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Set

Definition

A set is a well-defined collection of distinct objects, considered as an object in its own right.

Formally, a set \(S\) is determined by a property \(P\) such that for any object \(x\):

\[x \in S \iff P(x)\]

Notation

SymbolMeaning
\(x \in S\)\(x\) is an element of \(S\)
\(x \notin S\)\(x\) is not an element of \(S\)
\(\emptyset\)The empty set (contains no elements)
\(\{a, b, c\}\)The set containing \(a\), \(b\), and \(c\)
\(\{x : P(x)\}\)The set of all \(x\) satisfying property \(P\)

Properties

Well-definedness

For any object \(x\) and any set \(S\), exactly one of the following holds: \(x \in S\) or \(x \notin S\).

Distinctness

The elements of a set are distinct. \(\{1, 1, 2\} = \{1, 2\}\).

Unordered

The order of elements does not matter. \(\{1, 2, 3\} = \{3, 1, 2\}\).

Axioms (ZFC)

Modern set theory is built on the Zermelo-Fraenkel axioms with Choice (ZFC):

  1. Extensionality: Two sets are equal if they have the same elements.
  2. Empty Set: There exists a set with no elements.
  3. Pairing: For any two sets, there is a set containing exactly those two.
  4. Union: For any set of sets, there is a set containing all their elements.
  5. Power Set: For any set, there is a set of all its subsets.
  6. Infinity: There exists an infinite set.
  7. Separation: Given a set and a property, there is a subset satisfying that property.
  8. Replacement: The image of a set under a definable function is a set.
  9. Foundation: Every non-empty set contains an element disjoint from itself.
  10. Choice: Every set of non-empty sets has a choice function.

Theorem: Uniqueness of the Empty Set

Statement

There is exactly one empty set.

Proof

By the Empty Set axiom, there exists a set \(\emptyset\) with no elements.

Suppose \(E\) is another set with no elements. By Extensionality, for any \(x\): $\(x \in \emptyset \iff x \in E\)$

Since both are false for all \(x\), we have \(\emptyset = E\).

QED

Examples

Example 1

\[A = \{1, 2, 3\}\]

\(1 \in A\), \(4 \notin A\).

Example 2: Set-builder notation

\[B = \{n \in \mathbb{N} : n \text{ is even}\}\]

\(B\) is the set of all even natural numbers.

Example 3: The empty set

\[\emptyset = \{\}\]

Note: \(\emptyset \neq \{\emptyset\}\). The latter is a set containing one element (the empty set).

  • Element — Membership relation
  • Subset — Containment
  • Union — Combining sets
  • Intersection — Common elements
  • Cartesian Product — Pairs of elements