Biconditional statements

0:00- [Instructor] Started with this statement that we assumed to be true. If I am too close to a guinea pig, then I will sneeze. One way to think about this is if you view this first part of the statement, I am too close to a guinea pig, as statement p. and if you view the second part, I will sneeze, as p, one way to think about it is p is implying q here.

0:29And we could even draw this as a Venn diagram. So we could think about all of the times that I am too close to a guinea pig. So this is scenario p right over here, I am too close to a guinea pig, and all of these times I will sneeze, but there might be scenarios where I'll sneeze where I'm not too close to a guinea pig, at least the way it's said right here.

0:50So you can imagine that the set of all scenarios where I might sneeze might be bigger. So that would be q. So if I'm in p, I'm definitely in q, but if I'm in q, am I in p? Not necessarily. I may or may not be. And that's why in that previous video, when we looked at the converse of our assume statement, if I sneeze, so I'm swapping these around, if I sneeze, then I am too close to a guinea pig.

1:19We said that's not always going to be true because if I sneeze could be anywhere in this blue circle. You might be in this p in this yellow circle, or you might not. So we said this was false, but what if we wanted that to be true? What if we wanted the converse to be true?

1:37Well, that would be a situation where p implies q and q implies p. Q implies p. Getting the colors right. And so if I were to draw that as a Venn diagram, these circles would perfectly overlap. So this would be p, and then q would be the exact same circle. So the only scenarios in which I'm sneezing, I am too close to a guinea pig, and the only times that I'm too close to a guinea pig, I'm also sneezing.

2:14So that's q right over there. But how would I say that? This first conditional statement does not seem to be good enough. Well, there is a way to say that, or the way that's typically said that. Instead of saying if I am too close to a guinea pig, then I will sneeze, you would say, if and only if, only if p, I'm too close to a guinea pig, then q.

2:47So how is this different? Well, instead of just saying if, I'm saying only if, so there can't be any qs outside of the scenario p. There can't be any times I will sneeze outside of being close to a guinea pig because only if I am close to a guinea pig, then I will sneeze.

3:06And so if you want, you could pause this video or even after this video's done, you could see that if we had a by conditional statement right over here where q implies p and p implies q, and if we wanted to write that, what we would do is we would have statement p, we would have statement q.

3:23And in some logic classes, you'll see a bidirectional arrow here. P implies q and q implies p. From this statement, we could say if p, then q. But we could also say if q then p.