diff --git a/src/datamodel_code_generator/__main__.py b/src/datamodel_code_generator/__main__.py
index 73060d905..f21ce6654 100644
--- a/src/datamodel_code_generator/__main__.py
+++ b/src/datamodel_code_generator/__main__.py
@@ -608,8 +608,8 @@ def _init_preserved_type_origins() -> dict[type, str]:
     from collections.abc import Set as AbstractSet  # noqa: PLC0415
 
     return {
-        set: "Set",
-        frozenset: "FrozenSet",
+        set: "set",
+        frozenset: "frozenset",
         AbstractSet: "AbstractSet",
         ABCMutableSet: "MutableSet",
         ABCMapping: "Mapping",
diff --git a/src/datamodel_code_generator/parser/jsonschema.py b/src/datamodel_code_generator/parser/jsonschema.py
index e167d3544..ea2c9e92a 100644
--- a/src/datamodel_code_generator/parser/jsonschema.py
+++ b/src/datamodel_code_generator/parser/jsonschema.py
@@ -1287,7 +1287,9 @@ class decorator which does not preserve staticmethod descriptors.
 
         type_to_flag: dict[str, dict[str, bool]] = {
             "Set": {"is_set": True},
+            "set": {"is_set": True},
             "FrozenSet": {"is_frozen_set": True},
+            "frozenset": {"is_frozen_set": True},
             "Mapping": {"is_mapping": True},
             "MutableMapping": {"is_mapping": True},
             "Sequence": {"is_sequence": True},