MySQL 5.7

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create table tbl(
 jsn json
);

insert into tbl values
  ('{"1":["0", "3"],"2":["3","4"]}'),
  ('{"3":["1", "2"],"1":["3","4"]}'),
  ('{"2":["1", "2"],"4":["3","4"],"6":["3","4"]}'),
  ('{"2":["0", "3"],"3":["3","4"]}'),
  ('{"a":["0", "3"],"b":["3","4"],"c":["3","4"],"1":["3","4"]}'),
  ('{"1":["1", "3"],"2":["3","4"]}');
select jsn from tbl
;


jsn


{"1": ["0", "3"], "2": ["3", "4"]}

{"1": ["3", "4"], "3": ["1", "2"]}

{"2": ["1", "2"], "4": ["3", "4"], "6": ["3", "4"]}

{"2": ["0", "3"], "3": ["3", "4"]}

{"1": ["3", "4"], "a": ["0", "3"], "b": ["3", "4"], "c": ["3", "4"]}

{"1": ["1", "3"], "2": ["3", "4"]}


select *
  from tbl
  where JSON_CONTAINS(jsn, '{"1":["0"]}') = 1
     OR JSON_CONTAINS(jsn, '{"1":["3"]}') = 1


jsn


{"1": ["0", "3"], "2": ["3", "4"]}

{"1": ["3", "4"], "3": ["1", "2"]}

{"1": ["3", "4"], "a": ["0", "3"], "b": ["3", "4"], "c": ["3", "4"]}

{"1": ["1", "3"], "2": ["3", "4"]}