Conditional statements and logical equivalence
About this video
Starting from the true statement "If I am too close to a guinea pig, then I will sneeze," the video tests its converse, inverse and contrapositive. The converse and inverse can be false (you might sneeze from pepper or a cold), but the contrapositive is always true. It ends by linking this to the ideas of sufficient and necessary conditions.
You'll be able to
- form the converse, inverse and contrapositive of a conditional statement.
- decide whether each related statement must be true when the original is true.
- use a counterexample to show that a converse or inverse is false.
Key words
- conditional statement
- A statement of the form "If p, then q."
- converse
- The statement made by swapping the if-part and the then-part: "If q, then p."
- inverse
- The statement made by putting "not" in front of both parts: "If not p, then not q."
- contrapositive
- The statement made by swapping both parts and negating both: "If not q, then not p"; it is always true when the original is.
- sufficient
- Enough on its own to guarantee the result.
- necessary
- Must happen whenever the other thing happens, even if it does not guarantee it.
Transcript
- [Instructor] So let's assume the statement, "If I am too close to a guinea pig, then I will sneeze." Let's just assume that that is true. So based on assuming that is true, which of these statements are also going to be true all of the time? Pause this video and think about that before we do this together.
Okay, so this first statement, "If I sneeze, then I am too close to a guinea pig." Well, first of all, let's think about how this relates to the first one. The first one, "If I am too close to a guinea pig, then I will sneeze." In the second one, lemme do this.
Let me make sure I use my colors properly. In the second one over here, it says, "If I sneeze, then I am too close to a guinea pig." So here we've just swapped the, what we have after the if and what we have after the then. And just so you know what the fancy word for this is, it's useful to know.
Some people might say, "Oh, you're just saying the converse." Well, this is what the converse means. Well, you essentially switch the, what's behind or what's after the if and the then. But more interestingly, is this always going to be true If that first statement is true?
"If I'm too close to a guinea pig, then I will sneeze." Well, it's not saying that's the only reason you might sneeze. You might also sneeze because you inhaled some pepper or you have a cold or there's something else you're allergic to and it's not a guinea pig.
And all of those things might make you sneeze. So if you sneeze once again, maybe you inhaled some pepper, that doesn't necessarily mean you are too close to a guinea pig. There might not be a guinea pig anywhere near you. So this is not true, or we'll say this is false.
Now, let's look at this next statement. "If I am not too close to a guinea pig, then I will not sneeze." So first of all, let's just think about how this is different from our first statement. Here we have the, "I am too close to a guinea pig," but then we have the "not" there.
So it's negating it, "not." And then we also have the "not" instead of "I will sneeze." "I will not sneeze." So let me do, underline that just so you can see the structure. So we put a "not" in front of both of them. So this one, just so you know the fancy word for it, this is the inverse.
But more interestingly, again, is this going to be true? This is the inverse. Well, "If I'm not too close to a guinea pig, then I will not sneeze." Well, let's just use that example of not being anywhere close to a guinea pig, but you inhaled pepper again. Well, then you're going to sneeze, or that might be a reason that you sneeze even though you are not close, not too close to a guinea pig.
So this is not true. Just because you are not too close to a guinea pig does not guarantee that you'll not sneeze. Once again, I'm gonna say it again. You might be allergic to something else, or you might have a cold and there's not a guinea pig in sight. Alright, now, let's look at this last statement.
So this one has some nots in it, but it also swapped the if in the end, like the converse, but it also added the nots. So, "If I do not sneeze, then I am not too close to a guinea pig." Is this one true? Actually, before I even ask you that, what is this even called?
What's the fancy word for it? This is known as a contrapositive, very fancy word. It really just means that you swapped these two and you made them not. You negated them. So it's a combination of the inverse and the converse is the contrapositive. But is this true?
We're assuming that if you're close, if you're too close to a guinea pig, then you will sneeze every time. So if you're not sneezing, then there's no way that you're too close to a guinea pig. There might be other, you also might not be close to anything else that you're allergic to.
You also might not be inhaling salt, but you're definitely not too close to a guinea pig. Because if you were, then you would sneeze. So this one right over here, I would say, is definitely true. One way to think about it is, is that being too close to a guinea pig is sufficient for sneezing.
So if you just have this, then you are definitely going to sneeze. You don't need anything else. Now, going the other way, it isn't sufficient. So sneezing will be necessary when you are close to a guinea pig. But it's not sufficient to say that you are definitely close to a guinea pig.
Sneezing, as we said before, could be caused from something else. It could be caused from inhaling salt, et cetera, or pepper. We're inhaling all sorts of things in this video. So anyway, I'll leave you there. This is very important. These are important to be able to think this way, to think about logical statements and conditional statements this way.
It's also useful to know these words, converse, inverse, and contrapositive. People say, "Oh, you're just stating the contrapositive," et cetera, et cetera. And you say, what do they mean by contrapositive? They're just meaning you're switching the if in the end and you're putting a not for both of them.
Or if they're saying the inverse, that means you're just, it's the same statement, but it's the not for both of them or the converse, it means you are switching them.
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