Im trying to work out whether the following clause is satisfiable:

{x, y},{x,¬y},{¬x, y},{¬x,¬y},{x, z},{x,¬z},{y, z},{y,¬z}

My basic understanding is to work this out, you must give each literal a true or false assignment to work out if each clause is satisfiable so:

x=1, y=1, z=1,

{1,1}, {x,0}, {0,1}, {0,0}, {1,1}, {1,0}, {1,1}, {1,0}

Therefore the clause {x,y} and {x,z} are satisfiable?

I also heard you can switch the statement so:

x=1, y=0, z=1,

This means {x,y} is not satisfiable.

If we changed z to 0 as well {x,z} would not be satisfiable. What really confuses me here is when the truth assignments are changed, neither clauses are satisfiable.

Where am I going wrong?


1 Answer 1


You seem to assume that both literals in a clause must be true in order to satisfy the clause. However, it is sufficient that one of them is true. So with your first assignment all clauses but {¬x,¬y} are satisfied.

You also mix up some terminology:

  • A clause is a single group of literals like {x,y}. What you are trying to satisfy overall is a group of clauses or a formula.
  • When you are considering a specific assignment of values, you talk about clauses that are "satisfied", not "satisfiable". Satisfiability is a property of the formula as a whole. (A single clause is always satisfiable.)

PS: Since your formula only uses 3 variables it is feasible to try out each possible assignment of values in order to see if the formula is satisfiable.

  • 2
    $\begingroup$ Specifically, "satisfiable" means "can be made true by some assignment"; "satisfied" means "is made true by this assignment". $\endgroup$ Commented May 3, 2014 at 9:07

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