Set Theory Basics

Every time you screen a universe of stocks — "market cap over $1bn and momentum positive but not already in the portfolio" — you are doing set theory. You never call it that, but the words "and," "or," and "not" are precise operations on collections of objects, and probability theory borrows exactly this machinery to talk about outcomes. Before you can say "the probability of A or B," you need to know exactly what "A or B" is as a collection. That is what this page pins down.

The analogy: a filing cabinet of index cards

Picture a big filing cabinet holding one card for every stock in your universe. A set is simply any pile of cards you pull out and label — "large caps," "tech names," "stocks you currently own." Once you have two piles, three questions become natural: which cards are in either pile (push them together), which cards are in both piles (find the overlap), and which cards are in the whole cabinet but not in a given pile (everything left behind). Those three questions are union, intersection, and complement, and nothing more sophisticated is happening underneath the symbols.

The symbols, one at a time

Let $\Omega$ (a capital omega) denote the universe — the whole filing cabinet, every card that could possibly come up. A set $A$ is a sub-pile of cards drawn from $\Omega$, and $x \in A$ reads "the card $x$ is in pile $A$."

The three operations:

$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$

This is the union: every card that belongs to at least one of the two piles. In plain English, $A \cup B$ answers "large cap or tech."

$$A \cap B = \{x : x \in A \text{ and } x \in B\}$$

This is the intersection: only the cards that belong to both piles at once — the overlap. In plain English, $A \cap B$ answers "large cap and tech."

$$A^c = \{x \in \Omega : x \notin A\}$$

This is the complement: everything in the cabinet that is not in pile $A$. In plain English, $A^c$ answers "everything except large caps."

Two sets with no overlap at all — $A \cap B = \emptyset$, the empty pile — are called disjoint. That single idea is what lets probabilities be added instead of double-counted, and it is exactly the property Kolmogorov's third axiom relies on.

Worked example 1: screening a ten-stock universe

Say your universe $\Omega$ has ten stocks, numbered 1 through 10. Large caps are $A = \{1,2,3,4,5,6\}$. Positive-momentum names are $B = \{3,4,5,7,8\}$.

Union $A \cup B$: list every number appearing in either set, once each: $\{1,2,3,4,5,6,7,8\}$ — eight stocks pass "large cap or momentum."

Intersection $A \cap B$: numbers appearing in both lists: $\{3,4,5\}$ — three stocks are both large cap and positive momentum. Count them by hand: 3 is in A and in B, keep it; 4 is in both, keep it; 5 is in both, keep it; 1, 2, 6 are only in A, drop them; 7, 8 are only in B, drop them.

Complement $A^c$: everything in $\Omega$ not in $A$: $\{7,8,9,10\}$ — the small caps.

Worked example 2: counting via inclusion-exclusion

How many stocks are in $A \cup B$ without listing them? Count $A$: 6 stocks. Count $B$: 5 stocks. Add them: $6 + 5 = 11$. But stocks 3, 4, 5 got counted twice — once in each pile — so subtract the overlap once: $11 - |A \cap B| = 11 - 3 = 8$. That matches the union counted by hand above. The rule in general is

$$|A \cup B| = |A| + |B| - |A \cap B| ,$$

read as "add the two pile sizes, then remove the double-counted overlap." This exact formula is why $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ in every introductory probability course — probability is just a way of weighting each card instead of counting it as 1.

What this means in practice

A trading filter chain — liquidity screen and sector screen and not already-held — is a chain of intersections and complements applied to the universe $\Omega$ of tradable names. A probability question like "what's the chance the market is up or volatility is down" is a union of two events, computed with exactly the inclusion-exclusion rule above. Set notation is the plumbing every later probability concept sits on top of.