Set theory Key concepts

Set theory gives precise answers to “what objects are we talking about?” and “which objects are connected?”

Core concepts

• Element: $x\in A$ means $x$ belongs to Set $A$.
• Subset: $A\subseteq B$ means every element of $A$ is also in $B$.
• Empty set: arnothing contains no elements.
• Union/intersection: $A\cup B$ combines membership; $A\cap B$ keeps only common members.
• Difference/complement: $A\setminus B$ removes elements of $B$ from $A$.
• Cartesian product: $A\times B$ is the set of ordered pairs.
• Relation: any subset of a Cartesian product.
• Function: a relation with exactly one output for every input in the domain.

Standard sets

Number sets are the most common examples: $\mathbb{N}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R}\subseteq \mathbb{C}$. Intervals such as $[a,b]$ and $(a,b)$ are standard subsets of $\mathbb{R}$.

Applied intuition

In physics, a state space is a set of possible states; a trajectory is often a function from time into that set. In probability, events are subsets of a sample space. In ML, datasets are finite sets or sequences of examples.