CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two-event inclusion-exclusion: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Statement

For events $A$ and $B$ in a finite probability space, $P (A \cup B) = P (A) + P (B) - P (A \cap B) .$

Facts & Assumptions

Given: Events $A$ and $B$ in a finite probability space.

[L1]

If $C \subseteq D$ , then $P (D ∖ C) = P (D) - P (C)$ (Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space).

[L2]

Probability is additive on every finite pairwise-disjoint family of events (Probability is additive on every finite pairwise-disjoint family of events).

Proof

technique · direct

1.1

The sets $A$ and $B ∖ A$ are disjoint and have union $A \cup B$ , so $P (A \cup B) = P (A) + P (B ∖ A)$ .

1.2

Since $B ∖ A = B ∖ (A \cap B)$ , one has $P (B ∖ A) = P (B) - P (A \cap B)$ .

2.1

Substitution of step 1.2 into step 1.1 gives the formula.

step 1.1step 1.2algebra∎

Depends on

Used by

The union bound can be strict for overlapping events Example

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.4 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.3.3 (standard reference, not scraped)

Two-event inclusion-exclusion: P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B)