Matching columns in rows from two tables

I have the following tables:

-- all of the users
-- user_id are unique
CREATE TABLE users (
user_id BIGINT,
....
PRIMARY KEY (user_id)
);
-- all of the groups
-- user_id are unique
CREATE TABLE groups (
group_id BIGINT,
...
PRIMARY KEY (group_id)
);
-- all of the group sets
-- group_set_id are unique
CREATE TABLE group_sets (
group_set_id BIGINT,
...
PRIMARY KEY (group_set_id)
);
-- a user can be a member of one or more groups
CREATE TABLE membership (
user_id BIGINT,
group_id BIGINT,
....
FOREIGN KEY (user_id) REFERENCES users (user_id),
FOREIGN KEY (group_id) REFERENCES groups (group_id)
);

-- a group set is a set of one or more groups
CREATE TABLE group_sets (
group_id BIGINT,
group_set_id BIGINT,
....
FOREIGN KEY (group_id) REFERENCES groups (group_id),
FOREIGN KEY (group_set_id) REFERENCES group_sets (group_set_id)
);

I want to:

Find the group_set_id such that for a given user_id (input
parameter)
there is a one-to-one correspondence between the group_ids
associated
with the user_id and the group_ids associated with the group_set_id;

for every group_id that the user_id has, the group_set_id also has
it
and for every group_id that the group_set_id has, the user_id also
has it.
If there is no such group_set_id, then return null.

What query will generate the group_set_id?

There ought to be some combination of joins, intersections, etc. that
can generate the result
but I am getting hungup on the fact that the number of group_ids being
matched is not fixed.

Thanks for any help.

Richard