Original question and givens
The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. Does the ordered pair (14, 10) belong to R?
G001 — The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. Does the ordered pair (14, 10) belong to R?
Step-by-step derivation
S001 · implies
Before
source roles = [qf-relation-pair-membership, [A, B, R, operation, x, y]]
After
derived = [pair_membership, no-invented-domain]
Conditions: none
Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.
Step identifiers and dependencies
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
S002 · implies
Before
source roles = [relation A ORIGINAL, 100, [10, 11, 12]]
After
derived = [[10, 11, 12]]
Conditions: none
Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.
Step identifiers and dependencies
Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001
S003 · implies
Before
source roles = [relation B ORIGINAL, 100, [14, 15, 16]]
After
derived = [[14, 15, 16]]
Conditions: none
Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.
Step identifiers and dependencies
Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S002
S004 · implies
Before
source roles = [relation R ORIGINAL, 0, [10, 14]]
After
derived = [[10, 14]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003
S005 · implies
Before
source roles = [relation R ORIGINAL, 1, [11, 15]]
After
derived = [[11, 15]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004
S006 · implies
Before
source roles = [relation R ORIGINAL, 2, [12, 16]]
After
derived = [[12, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005
S007 · implies
Before
source roles = [[10, 11, 12], [14, 15, 16], [[10, 14], [11, 15], [12, 16]]]
After
derived = [Yes]
Conditions: none
FIRST qualify the WHOLE R as a subset of A×B.A relation may be one-to-many/many-to-one and need not be total or a function.Typed raw duplicate pairs may normalize.
Step identifiers and dependencies
Rule: Relation_WHOLE_R_subset_declared_AcrossB_NO_function_premise · Uses: G001, S006
S008 · implies
Before
source roles = [14, 10]
After
derived = [[[[10, 14], No], [[11, 15], No], [[12, 16], No]], No]
Conditions: none
Compare the COMPLETE ordered pair,not separately occurring coordinates.A query outside carriers or in empty R is a LEGAL No.Do not force queried labels into A/B or confuse(y,x).
Step identifiers and dependencies
Rule: Relation_PAIR_query_keep_component_order_OUTSIDE_is_legal_No · Uses: G001, S007