MySQL 8.0

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CREATE TABLE df (
    student int,
    var1 varchar(100),
    start date,
    end date
);
INSERT INTO df (student, var1, start, end) VALUES
(1, 'a', '2010-01-01 00:00:00', '2013-01-01 00:00:00'),
(1, 'b', '2010-01-01 00:00:00', '2013-01-01 00:00:00'),
(1, 'b', '2013-05-05 00:00:00', '2015-09-09 00:00:00'),
(1, 'a', '2017-10-10 00:00:00', '2018-09-01 00:00:00'),
(2, 'c', '2010-01-01 00:00:00', '2014-01-01 00:00:00'),
(2, 'a', '2015-01-01 00:00:00', '2017-09-01 00:00:00'),
(2, 'b', '2019-01-01 00:00:00', '2023-03-05 00:00:00');

Records: 7  Duplicates: 0  Warnings: 0


with yearlist as (
    select '2010-03-01' as start_date, '2011-03-01' as end_date union all
    select '2011-03-01', '2012-03-01' union all
    select '2012-03-01', '2013-03-01' union all
    select '2013-03-01', '2014-03-01' union all
    select '2014-03-01', '2015-03-01' union all
    select '2015-03-01', '2016-03-01' union all
    select '2016-03-01', '2017-03-01' union all
    select '2017-03-01', '2018-03-01' union all
    select '2018-03-01', '2019-03-01' union all
    select '2019-03-01', '2020-03-01'
), studentlist as (
    select distinct student
    from df
)
select
    studentlist.student,
    yearlist.start_date,
    yearlist.end_date,
    case when exists (
        select *
        from df
        where df.var1 = 'a'
        and df.student = studentlist.student
        and df.end > yearlist.start_date
        and df.start < yearlist.end_date
    ) then 'true' else 'false' end as has_a
from yearlist
cross join studentlist
order by studentlist.student, yearlist.start_date


student
start_date
end_date
has_a


1
2010-03-01
2011-03-01
true

1
2011-03-01
2012-03-01
true

1
2012-03-01
2013-03-01
true

1
2013-03-01
2014-03-01
false

1
2014-03-01
2015-03-01
false

1
2015-03-01
2016-03-01
false

1
2016-03-01
2017-03-01
false

1
2017-03-01
2018-03-01
true

1
2018-03-01
2019-03-01
true

1
2019-03-01
2020-03-01
false

2
2010-03-01
2011-03-01
false

2
2011-03-01
2012-03-01
false

2
2012-03-01
2013-03-01
false

2
2013-03-01
2014-03-01
false

2
2014-03-01
2015-03-01
true

2
2015-03-01
2016-03-01
true

2
2016-03-01
2017-03-01
true

2
2017-03-01
2018-03-01
true

2
2018-03-01
2019-03-01
false

2
2019-03-01
2020-03-01
false

student	start_date	end_date	has_a
1	2010-03-01	2011-03-01	true
1	2011-03-01	2012-03-01	true
1	2012-03-01	2013-03-01	true
1	2013-03-01	2014-03-01	false
1	2014-03-01	2015-03-01	false
1	2015-03-01	2016-03-01	false
1	2016-03-01	2017-03-01	false
1	2017-03-01	2018-03-01	true
1	2018-03-01	2019-03-01	true
1	2019-03-01	2020-03-01	false
2	2010-03-01	2011-03-01	false
2	2011-03-01	2012-03-01	false
2	2012-03-01	2013-03-01	false
2	2013-03-01	2014-03-01	false
2	2014-03-01	2015-03-01	true
2	2015-03-01	2016-03-01	true
2	2016-03-01	2017-03-01	true
2	2017-03-01	2018-03-01	true
2	2018-03-01	2019-03-01	false
2	2019-03-01	2020-03-01	false