Original question and givens
The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. For the explicit C={12, 13}⊆A, list R[C] in increasing order with no duplicates.
G001 — The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. For the explicit C={12, 13}⊆A, list R[C] in increasing order with no duplicates.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-relation-subset-image, [A, B, C, R, operation]]
After
derived = [subset_image, 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, [11, 12, 13]]
After
derived = [[11, 12, 13]]
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, [15, 16, 17]]
After
derived = [[15, 16, 17]]
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, [12, 15]]
After
derived = [[12, 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, S003
S005 · implies
Before
source roles = [relation R ORIGINAL, 1, [13, 16]]
After
derived = [[13, 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, S004
S006 · implies
Before
source roles = [[11, 12, 13], [15, 16, 17], [[12, 15], [13, 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, S005
S007 · implies
Before
source roles = [relation image C ORIGINAL, 100, [12, 13]]
After
derived = [[12, 13]]
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, S006
S008 · implies
Before
source roles = [12, 15, [12, 13]]
After
derived = [Yes]
Conditions: none
For EACH pair,test whether its FIRST coordinate belongs to C and collect ALL qualifying SECOND coordinates.Do not select only one output or substitute C itself.
Step identifiers and dependencies
Rule: Relation_IMAGE_EACH_pair_SOURCE_membership_then_ALL_outputs · Uses: G001, S007
S009 · implies
Before
source roles = [13, 16, [12, 13]]
After
derived = [Yes]
Conditions: none
For EACH pair,test whether its FIRST coordinate belongs to C and collect ALL qualifying SECOND coordinates.Do not select only one output or substitute C itself.
Step identifiers and dependencies
Rule: Relation_IMAGE_EACH_pair_SOURCE_membership_then_ALL_outputs · Uses: G001, S008
S010 · implies
Before
source roles = [[11, 12, 13], [15, 16, 17], [12, 13], [[12, 15], [13, 16]]]
After
derived = [[15, 16], {15, 16}]Conditions: none
FIRST require C⊆A.The complete image R[C]lies in B and is deduplicated/numerically sorted.Empty C or no selected edges gives an empty image,not a claim about other subsets.
Step identifiers and dependencies
Rule: Relation_IMAGE_complete_selected_pairs_dedup_numeric_output · Uses: G001, S009