Original question and givens
The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. For the explicit C={4, 5}⊆A, list R[C] in increasing order with no duplicates.
G001 — The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. For the explicit C={4, 5}⊆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, [3, 4, 5]]
After
derived = [[3, 4, 5]]
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, [7, 8, 9]]
After
derived = [[7, 8, 9]]
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, [3, 8]]
After
derived = [[3, 8]]
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, [4, 8]]
After
derived = [[4, 8]]
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 = [[3, 4, 5], [7, 8, 9], [[3, 8], [4, 8]]]
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, [4, 5]]
After
derived = [[4, 5]]
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 = [3, 8, [4, 5]]
After
derived = [No]
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 = [4, 8, [4, 5]]
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 = [[3, 4, 5], [7, 8, 9], [4, 5], [[4, 8]]]
After
derived = [[8], {8}]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