The first one is satisfiable.

The next one is satisfiable.

P might be an empty relation, so this is satisfiable.

And a satisfiable sentence is one that is true in some models, but not necessarily in all the models.

I have just given you an example for making it true or making it false. Hence it is satisfiable.

I turn that into a 3 colorability problem that can be colored if and only if the formula is satisfiable.

So this graph that we built is colorable if and only the entire formula is satisfiable. That's kind of cool.

So what I want you to do is tell me for each of these sentences, whether it is valid, satisfiable but not valid, or unsatisfiable, in other words, false for all models.

It is at least satisfiable, but it is not valid, because if I flip B to false then the expression on the left becomes false, but the right expression is not affected.

I can quickly turn it into a 3 colorability problem, which is going to be a graph such that that graph is 3 colorable if and only if the original formula the 3 CNF formula that were given is satisfiable.

A gadget is going to be a little chunk of a graph but it's going to have very particular property to it that it's going to be three colorable in the context to the rest of the graph if and only if the corresponding clause is satisfiable in the context of the assignment.

What we're going to do is we're going to create a ??? formula that is satisfiable if and only if this graph is 3 colorable, so that it's really the same problem, and the way that we're going to do that is we're going to create a bunch of boolean variables, 12 of them to be exact, corresponding to each of the four nodes a, b, c, and d, and for each node, we'll a boolean variable saying whether that node is red or green or yellow, the three colors.