Set Theory/Set Operations

Subset

We can define the subset relation on two sets A and B as A is a subset of B if for every x, $x\in A$ implies $x\in B$. We denote this relation $A\subseteq B$. If we also have the fact that $A\ne B$, then we call A a proper subset of B, and write $A\subset B$.

Operations

We can define the union of two sets A and B using the axioms of union and pair. By axiom of pair {A,B} is a set, so define by axiom of union $A\cup B=\cup \{A,B\}$.

We define the intersection of two sets using the axiom of separation. Let $P(x)$ hold if and only if $x\in B$, then define $A\cap B=\{x\mid x\in A\wedge P(x)\}$.

We define the difference of two sets using the axiom of separation. Let $P(x)$ hold if and only if $x\in ̸B$, then define $A\setminus B=\{x\mid x\in A\wedge P(x)\}$.