in this video we're going to talk about how to write the converse the inverse and the contrapositive of a conditional statement so what exactly is a conditional statement what do you think it is now you've seen this before and the simplest way to describe it is as follows if p then q that is a conditional statement so let's say if you live in los angeles then you live in california that's an example of a conditional statement the part that comes after the if which is denoted by p that is the hypothesis the stuff that comes after q is known as the conclusion now you need to be familiar with the symbols that is associated with negation so this symbol means not p so for example let's say it is sunny outside the negation of that statement is it is not sunny outside so if you see this symbol it just means it's not happening it's a negation now what about the converse what is the converse of the original conditional statement the converse is basically the reverse of the conditional statement so you just gotta switch q and p so if q then p that is the converse if the conditional statement is true and if the converse is true what you have is a biconditional statement but the converse is not always true sometimes it's true sometimes it's not the next term you need to be familiar with is the inverse now the inverse is simply the negation of the conditional statement if not p then not q so that's the inverse the last one needs to be familiar with is the contrapositive the contrapositive is basically the reverse negation of the conditional statement if not q then not p and i'll give you some examples of that and finally we talked about the biconditional statement in a biconditional statement what you have is it occurs when the conditional statement and the converse have the same true value so if the conditional statement is true and if the converse is true then you have a biconditional statement in the conditional statement if p then q can be symbolized by this expression in a biconditional statement it can go both ways so if p then q or if q then p both will be true or both will be false that's a biconditional statement so let's try an example if you live in los angeles then you live in california so this is the conditional statement now what is the hypothesis and what is the conclusion the hypothesis is the stuff that comes after the if part so the hypothesis is living in los angeles the conclusion is living in california now how can we write the converse and the inverse and in addition the contrapositive of the statement so right now keep in mind the hypothesis is p so p is living in los angeles and q is living in california so right now we have if p then q so that's the conditional statement to write the converse we need to recall that the converse is the reverse of the conditional statement if q then p so what we need to do is switch the second part of the sentence with the first part so therefore the converse will be as follows if you live in california then you live in los angeles so that is the converse now is the converse a true statement or is it a false statement right now we're assuming that the conditional statement is true but the converse is a false statement someone who lives in california doesn't have to live in los angeles they could live in sacramento they could live in san francisco or san diego so the converse is false so if you have a true conditional statement and a false converse statement then this is not a biconditional statement if the converse was true and if the conditional statement assuming that's true as well then we would have a biconditional statement now let's write the inverse of the conditional statement so recall that the inverse is simply the negation of the condition the conditional statement excuse me so if not p then not q so the sentence will be as follows if you don't live in los angeles i'm going to write la for los angeles then you don't live in california so i'm going to write ca for california so that's the inverse if you don't live in los angeles you don't live in california so you don't reverse the sentence you simply negate the sentence from being yes to know now what about the contrapositive oh by the way is the inverse true or false if you don't live in l.a then you don't live in california what would you say well just because you don't live in l.a doesn't mean you don't live in california because you can live in san francisco which is not in l.a but still within the state of california so in this case the inverse is false it turns out that if the converse is false the inverse is going to be false those two have the same true value if the converse is true then the inverse is going to be true as well now let's write the contrapositive statement so remember the contrapositive is basically the reverse negation of the conditional statement and it says if not q then not p so we got to reverse it and negate it so the sentence will be as follows if you don't live in california then you don't live in los angeles now is that a true statement it turns out that it is because if you don't live in a state of california you can't live inside a city within california which is la now it turns out that if the conditional statement is true then the contra positive will be true and if the conditional statement is false the contrapositive will be false so the conditional statement and the contrapositive will always have the same truth value they will always be the same in the same way as the converse and the inverse are always the same any time you have a reverse negation of something it's like a double reverse it's going to take you back to the original value and that's what's happening with the contrapositive not only do you reverse the statements but you negate it as well which makes it true again go ahead and work on this example problem if i am hungry then i will eat pizza so given that conditional statement write the converse the inverse and the contrapositive so let's start with the converse the converse is simply the reverse of the conditional statement so we could say if i eat pizza then i am hungry so that's the converse we just got to reverse the sentence now let's move on to the inverse the inverse is simply a negation of the sentence so if i am not hungry then i will not eat pizza so that's the inverse you just gotta negate the original sentence and finally let's write the contrapositive which is a reverse negation of the original sentence so not only do we have to reverse it we need to negate it as well so if if i don't eat pizza or if i will not eat pizza however you want to phrase it if i don't eat pizza then i am not hungry so that's the contrapositive you