Alphabeta MathHow 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.
Sets with
Statement refuted
Refuted claim: for all sets , , , . The witness is and : the pair mixes a first coordinate from with a second coordinate from , so it lies in the right-hand side and in neither product on the left.
Facts & Assumptions
Given: and .
holds if and only if for some and some (The Cartesian product ).
if and only if and ( if and only if and ).
holds if and only if or (, , , , and ).
is the set whose elements are exactly and , and (The unordered pair and the singleton ).
There is exactly one set with no elements (There is exactly one set with no elements, written ).
means that every element of is an element of (Subset , proper subset , and the separation notation ).
Counterexample
, since the second has an element and the first has none; so and , while and .
The inclusion from left to right always holds: an element of is a pair with and , hence with and , so it lies in ; the same argument applies to .
The pair lies in , because gives and gives .
It lies in neither product on the left. Membership in would give , and membership in would give ; the characterising property makes the coordinates unambiguous, and step 1.1 rules out both.
The inclusion of step 1.2 therefore omits the element exhibited at step 2.1, so it is proper.
Depends on
- The Cartesian product $A \times B := \{\, z \in \mathcal{P}(\mathcal{P}(A \cup B)) : \exists a \in A\ \exists b \in B\ z = (a,b) \,\}$
- The union $\bigcup x$ of a set, and the binary union $a \cup b := \bigcup \{a,b\}$
- $\bigcup \varnothing = \varnothing$, $\bigcup \{a\} = a$, $\bigcup \{a,b\} = a \cup b$, $\bigcap \{a\} = a$, and $\bigcap \{a,b\} = a \cap b$
- The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$
- There is exactly one set with no elements, written $\varnothing$
- $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$
- Subset $x \subseteq y$, proper subset $x \subsetneq y$, and the separation notation $\{\, z \in x : \varphi(z) \,\}$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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
- Cartesian product (Wikipedia) (standard reference, not scraped)
- Algebra of sets (Wikipedia) (standard reference, not scraped)
- B. Kaya, MATH 320 Set Theory (METU), §2.1 (standard reference, not scraped)