 we're going to be talking about truth tables. Truth tables give us a way to show the results of an expression for all of the possible inputs it could have. There are lots of different ways of writing a truth table. We'll show one simple way is, have say, all of our propositions on the left hand side. And then we'll have some combinations on the right hand side here. So if I want to do p and q, well, I have two different options for p and for q. So I will write down all of those possible options. So those are all of the four possible combinations of p and q that I can have. For the AND operation, I know that the AND is only going to be true if both p and q are true. So this first case where p and q are true, I will write down true. The second case, p is true but q is false. So this is false. If p is false and q is true, well, that's still false. And if both p and q are false, then AND of both of those is false. I can do this as well for p or q. So p is q and q is true, so p or q is true. p is true but q is false, so p or q will still be true. When p is false and q is true, well, q is still true, so the or of both of those is true. But when both p and q are false, the or is false. So I can build these for any of the obvious expressions that I have. I can also use this when working with more complex expressions. So I can start combining expressions. So here I have p and q or p. So in this case, one of my inputs will be the p and q that I've got over here, and the other input to this or is p. So this expression will be true in any case where either p and q is true or p is true. I've got all the possible values for p as well as all the possible values for p and q. So p is true and p and q is true. Well, that will mean that the or of both of those is true. Here p is true but p and q is false. The or will still be true. For here I've got p is false and p and q is false. So that will be false. Or I've got false or false is false. So I can build these up and make lots of complex propositions and express what their actual outputs are just by finding some building blocks. It can also help me to show when two expressions are equivalent. So two expressions will be logically equivalent when for all of their possible inputs, the outputs match. So in this case, this TTFF turns out it matches our inputs for just p. So this expression is logically equivalent to just p.