[Music] hello everyone and welcome to another math tutorial in this video we're going to begin a new series of videos in a unit titled reasoning and proofs and this video and actually the first couple of videos are just really going to lay the foundation or the framework for what we need to build up to writing geometric proofs this first section is called conditional statements and so let's begin right there with just a definition of what is a conditional statement so a conditional statement is a logical statement that has two parts it's going to have a hypothesis and for the purposes of shorthand we're going to call the hypothesis p and it's also going to have a conclusion and we'll call the conclusion q okay so that p and q uh labeling is going to be used pretty often within this section and just maybe the first couple of videos in this playlist so when written in if then form the if part contains the hypothesis and the then part is going to contain the conclusion so in shorthand we're writing this in the form if p then q okay this first example asks us to rewrite the conditional and if then form so we have a conditional statement here uh it says all members of the soccer team have practice today the issue with this is it is not written in this if then form so we want to rewrite this in if-then form so i might say if somebody is a member of the soccer team then they will have practice today so let's see what that looks like okay so again if someone is a member of the soccer team then they have practice today so you can see the same information is there in this conditional statement members of the soccer team we have that they're going to have practice today we've got that we've just split it up we've got this if then form where we've got the hypothesis someone is a member of the soccer team followed by the conclusion that they are going to have practice today next we're going to build up to what we call some related conditionals and the first thing that we need in order to build up into these related conditional statements is the idea of negation now in logic negation is simply just the opposite of the original statement and as far as notation goes symbolically you might see this expressed like that if i saw that this is going to mean not p just that symbol right there so the negation is again it's just the opposite and we're going to do a couple example problems of writing the opposite all right the directions on this slide are to write the negation of the statement so the first statement is that the ball is blue if i want to write the negation of this statement generally in writing the negation we're going to include the word not okay we're just going to add that in okay so if the ball is blue the opposite of that is the ball is not blue what we don't want to do is try and write the opposite by like changing the color i hear that sometimes somebody's going to want to say well the opposite of the ball is blue is to say the ball is red and that's not true we want to do this by just simply including the word not now sometimes and this this next statement illustrates this is that the word not has already been included so if i want to write the opposite of the dog is not white i'm simply going to write the dog is white so in this case i've taken the word not out of the statement and so it's simply a matter of including or excluding the word not and we can very simply state the opposite or negation of our statement okay next i want to talk about our related conditionals and to do this we're starting under the assumption that we have our if p then q so let's just write this at the top we're starting with okay if p then q which could be written in symbols like that okay so this is the shorthand symbolic notation for if p then q which is p the arrow and then q so i'm going to do some of this shorthand notation as i write the converse inverse and contrapositive so let's begin with the converse the converse to write the converse you simply will switch the hypothesis and conclusion so instead of writing if p then q you're going to write if q then p okay so that's the first related conditional it is called the converse the next related conditional is the inverse and when you're writing the inverse you are going to negate or state the opposite of the hypothesis and conclusion so you're not going to switch them like up here you're going to keep them in the exact same order as the original but you're just going to make them both the opposite so symbolically that looks like this if not p then not q okay the last one is going to be called the contrapositive and in the contrapositive you are going to so you're going to switch the hypothesis and conclusion statements and you're also going to negate them both so you can think of the contrapositive as like a blending of the converse and the inverse so symbolically it's going to look like this if not q then not p okay so those are our four conditional statements we've got the original if p then q and the converse if q then p we have the inverse if not p then not q and we have the contrapositive if not q then not p all right let's do a little bit of practice writing all four of those conditional statements all right to get this problem started the example says to let p be the statement you are a guitar player and let q be the statement you are a musician we're going to write each statement and then determine if the statement that we wrote is true or false okay the first statement we're going to write is our conditional statement and the conditional statement again is just simply going to take the form if p then q okay so we're going to write that statement using guitar player and musician up here so here's what that would look like okay so i wrote the statement if you are a guitar player then you are a musician so now the question is is that a true statement or is that a false statement well i would say that that is certainly a true statement if you are playing any musical instrument then i'm going to say that you fall probably fall under the category of musician okay next we're going to write the converse and the converse statement we are going to take the q and the p are hypothesis and conclusion and we're going to switch them so now we're going to write if q then p so that's going to look like this okay so switching the p and the q i've now written the statement if you are a musician then you are a guitar player and the question again is is that a true or a false statement well it might be true it might be true that you're a musician and you also play guitar however it's not always going to be true i could be a piano player and still be a musician so i can't say that this converse statement is going to necessarily be true all the time all right for our next statement we want to write the inverse and recall that the inverse is when we negate both hypothesis and conclusion so i'm going to write if not p then not q not going to do any switching of the p and the q i keep them in the same order as the original i'm just going to make them both opposites so i might write this statement okay so i have if you are not a guitar player then you are not a musician okay so now the question is is that a true statement or a false statement i would make the argument that that is a false statement okay if you are not a guitar player you might still be a musician because maybe you play another instrument so that's got to be a false statement as it's not always going to be true alright finally let's write the contrapositive the contrapositive again is going to switch the p and the q and it's also going to negate both statements so i'm going to write if not q then not p and so writing that out with my statements is going to look like this if you are not a musician then you are not a guitar player okay is that true or false well i'm going to say that's true if you're not a musician then that means you don't play anything so if you're not a musician you're definitely not a guitar player so i'm going to call that a true statement okay this next example problem is just like the previous one in that i'm going to ask you to write the conditional statement the converse the inverse and the contrapositive and you're going to determine if those are true statements or false statements and i think it'd be a good idea just to practice if you go ahead and hit pause on the video and try and write all four statements without my help and then when you've written the statements and you've determined true or false then press play and watch it play out and i'll go ahead and speed up the playback as well of the video so you don't have to watch me write very slow along the way so go ahead and hit pause give these four statements a chance on your own and then when you're ready hit play and check your answers okay there's my first two answers for the conditional and the converse next up let's show you the inverse and contrapositive okay there are my two statements for the inverse and the contrapositive okay the final thing we're going to talk about in this video is a biconditional statement a biconditional statement is one that can replace the conditional and the converse when both the conditional and the converse statements are true statements okay so you can write this i'm going to underline this when both the conditional and its converse are true and when that's the case you can replace the if then with if and only if so when you read something and you read this if and only if that you can imply then that the conditional and its converse are both true statements let's see an example of that all right our final example of the video to explore this biconditional statement we've got the example that we're going to let p be that two lines intersect to form a right angle and q be the statement they are perpendicular lines so we're asked to write the following we're going to write the conditional the converse and then this biconditional statement so i'm going to begin with the conditional all right the statement is if two lines intersect to form a right angle then they are perpendicular lines that ladies and gentlemen is a true statement okay next let's take a look at the converse all right the converse statement reads if two lines are perpendicular lines then they intersect to form a right angle and again that is a true statement okay so because both the conditional and the converse statements are true statements we are allowed to combine those two statements into one biconditional statement and the biconditional statement might look something like this okay so i have the statement two lines intersect to form a right angle if and only if they are perpendicular lines so you can see how i've combined these two statements with my biconditional i still led with my original hypothesis and i kept the conclusion at the end of the statement but we've just kind of merged these two ideas these two statements that were both true statements we've just been able to kind of merge or blend them into a single statement and preserve the meaning the validity of both statements in the process all right that concludes our video on conditional statements i hope you found this video helpful if you have any comments or questions i encourage and invite you to leave them in the comments section below um if you found this video helpful please support the channel by giving it a thumbs up and i'll see you in the next video hmm