MySQL 8.0

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CREATE TABLE myTable (
  `ltiNum` INTEGER,
  `ltiCd` VARCHAR(1),
  `updateDate` VARCHAR(10),
  `closedOrNot` VARCHAR(8)
);

INSERT INTO myTable
  (`ltiNum`, `ltiCd`, `updateDate`, `closedOrNot`)
VALUES
  ('1', 'A', '2020-07-01', 'NULL'),
  ('2', 'B', '2020-07-01', 'Closed'),
  ('3', '0', '2020-07-01', 'NULL'),
  ('4', 'D', '2020-07-01', 'NULL'),
  ('5', 'E', '2020-07-01', 'NULL'),
  ('1', 'A', '2020-08-01', 'existing'),
  ('3', '0', '2020-08-01', 'existing'),
  ('4', 'D', '2020-08-01', 'existing'),
  ('5', 'E', '2020-08-01', 'existing'),
  ('6', 'F', '2020-08-01', 'New');
update myTable 
set closedOrNot = 
        (case when (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 where updateDate Like '2020-08%'
                    and CONCAT(ltiNum,ltiCd) in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-07%'       
        )) then 'existing'
        when (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 where updateDate Like '2020-08%'
                    and CONCAT(ltiNum,ltiCd) not in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-07%'
        )) then 'New'        
         when (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 where updateDate Like '2020-07%'
                    and CONCAT(ltiNum,ltiCd) not in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-08%'
        )) then 'Closed'
    end);
Subquery returns more than 1 row

 SELECT CONCAT(ltiNum,ltiCd) 
 FROM (SELECT * FROM myTable) t1 
 where updateDate Like '2020-08%'
                    and CONCAT(ltiNum,ltiCd) in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-07%' )      


CONCAT(ltiNum,ltiCd)


1A

30

4D

5E


SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 where updateDate Like '2020-08%'
                    and CONCAT(ltiNum,ltiCd) not in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-07%'
        )


CONCAT(ltiNum,ltiCd)


6F


SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 where updateDate Like '2020-07%'
                    and CONCAT(ltiNum,ltiCd) not in (SELECT CONCAT(ltiNum,ltiCd) FROM (SELECT * FROM myTable) t1 WHERE updateDate Like '2020-08%'
        )


CONCAT(ltiNum,ltiCd)


2B