Set Theory

Zermelo-Fraenkel set theory with axiom of choice (ZFC) — 9 axioms all together — is being used in this module.

Definitions

• $x \in A^\text{c} \iff x \not\in A$
• $x \in A\cup B \iff x \in A \lor x \in B$
• $x \in A\cap B \iff x \in A \land x \in B$
• $A \subset B = \forall x (x \in A \implies x \in B)$
• $A - B = A \cap B^\text{c}$

Required Proofs

• $(A\cap B)^\text{c} = A^\text{c} \cup B^\text{c}$
• $(A\cup B)^\text{c} = A^\text{c} \cap B^\text{c}$
• $A \cap (B \cup C) = (A \cap B)\cup (A\cap C)$
• $A \cup (B \cap C) = (A \cup B)\cap (A\cup C)$
• $A \subset A\cup B$
• $A\cap B \subset A$

The Axioms

All the axioms defined in Zermelo-Fraenkel set theory and axiom of choice are mentioned here for the sake of completeness. Their exact, formal definition is not included here. Formal definitions can be found on ZFC set theory - Wikipedia.

Axiom of Extensionality

Two sets are equal (are the same set) if they have the same elements.

Axiom of Regularity

A set cannot be an element of itself.

Axiom of Specification

Subsets that are constructed using set builder notation, always exists.

Axiom of Pairing

If $x$ and $y$ are sets, then there exists a set which contains both $x$ and $y$ as elements.

Axiom of Union

The union of the elements of a set exists.

Axiom Schema of Replacement

The image of a set under a definable function will also be a set.

Axiom of Infinity

There exists a set having infinitely many elements.

Axiom of Power Set

For any set $x$, there exists a set $y$ that contains every subset of $x$:

Axiom of Well-Ordering (Choice)

I don’t understand this axiom. If you do, let me know.