First of all: how is the operator “{}” really called? The documentation ist not very good at answering this.

I call it powerset operator because it creates a powerset of all elements.

{ z(1) ; z(2)}.

Creates the output:

$ clingo 001.lp 0
clingo version 5.4.1
Reading from 001.lp
Solving...
Answer: 1

Answer: 2
z(2)
Answer: 3
z(1)
Answer: 4
z(1) z(2)
SATISFIABLE

Models       : 4
Calls        : 1
Time         : 0.000s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s

When I constrict the cardinality to 0, 1 or 2 elements by writing:

{ z(1) ; z(2)} = cardinality.

then I get the output

$ clingo 002.lp -c cardinality=0 0
clingo version 5.4.1
Reading from 002.lp
Solving...
Answer: 1

SATISFIABLE

Models       : 1
Calls        : 1
Time         : 0.000s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s


$ clingo 002.lp -c cardinality=1 0
clingo version 5.4.1
Reading from 002.lp
Solving...
Answer: 1
z(1)
Answer: 2
z(2)
SATISFIABLE

Models       : 2
Calls        : 1
Time         : 0.000s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s

$ clingo 002.lp -c cardinality=2 0
clingo version 5.4.1
Reading from 002.lp
Solving...
Answer: 1
z(2) z(1)
SATISFIABLE

Models       : 1
Calls        : 1
Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.001s

This is all easy understandable so far.

Lets go to a different notation with a similar result:

{ z(Y) } :- Y=(1;2).

$ clingo 003.lp 0
clingo version 5.4.1
Reading from 003.lp
Solving...
Answer: 1

Answer: 2
z(1)
Answer: 3
z(2)
Answer: 4
z(2) z(1)
SATISFIABLE

Models       : 4
Calls        : 1
Time         : 0.000s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s

But when I constrain this construct

{ z(Y) } = cardinality :- Y=(1;2).

then I get:

$ clingo 004.lp -c cardinality=0 0
clingo version 5.4.1
Reading from 004.lp
Solving...
Answer: 1

SATISFIABLE

Models       : 1
Calls        : 1
Time         : 0.000s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.000s

$ clingo 004.lp -c cardinality=1 0
clingo version 5.4.1
Reading from 004.lp
Solving...
Answer: 1
z(2) z(1)
SATISFIABLE

Models       : 1
Calls        : 1
Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.001s

$ clingo 004.lp -c cardinality=2 0
clingo version 5.4.1
Reading from 004.lp
Solving...
UNSATISFIABLE

Models       : 0
Calls        : 1
Time         : 0.001s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)
CPU Time     : 0.001s

I have at this point no idea what actually happens and what it is good for.

Thanks in advance for your answers.