LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 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.

Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space

Statement

In a finite probability space $(Ω, w)$ , events $A \subseteq B \subseteq Ω$ satisfy $0 \leq P (A) \leq P (B) \leq 1,$ and $P (\emptyset) = 0, P (Ω) = 1, P (A^{c}) = 1 - P (A), P (B ∖ A) = P (B) - P (A) .$ Probability zero need not imply that an event is empty.

Facts & Assumptions

Given: A finite probability space $(Ω, w)$ and events $A \subseteq B$ .

[L1]

Event probability is the sum of the nonnegative weights of its outcomes, and the sum of all outcome weights is $1$ (Finite probability spaces, outcome weights, events, and event probabilities).

[L2]

Finite sums preserve nonnegativity and order (Laws of finite sums and finite products).

[L3]

The real numbers form a totally ordered field (The reals form a totally ordered field).

Proof

technique · direct

1.1

The empty sum is $0$ , while the sum over $Ω$ is $1$ , so $P (\emptyset) = 0$ and $P (Ω) = 1$ .

1.2

Since every summand in $P (A)$ is nonnegative, $0 \leq P (A)$ . Splitting the sum over $B$ into $A$ and $B ∖ A$ gives $P (B) = P (A) + P (B ∖ A)$ , so $P (A) \leq P (B)$ .

L1L2

2.1

Taking $B = Ω$ in step 1.2 gives $P (A^{c}) = 1 - P (A)$ ; rearranging the same identity for general $A \subseteq B$ gives $P (B ∖ A) = P (B) - P (A)$ .

step 1.1step 1.2L3algebra

3.1

The definition permits zero weights, so a singleton outcome of weight zero is a nonempty event of probability zero. All displayed conclusions follow.

L1step 1.1step 1.2step 2.1∎

Depends on

Used by

Dependency tree · two levels

21 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 1.1 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.3.2-1.3.3 (standard reference, not scraped)