it’s time to talk about groups what is a group a group is a very important concept in abstract algebra we won’t be going into too much details with groups there’s a lot of theorems that are left purely to mathematicians but it’s good for linguists to know what a group is so a group g this nice fancy curse of g consists of a set g and operation which we call circle and a group satisfies four axioms so it’s a pair with a set and an operation we’ve seen this before but there are four things that must occur in a group first of all g is closed under the operation which means that if i have two elements a b in our set g then it better be the case that a dot b is also going to be in g the operation is going to be associative we already know the associativity is i’m not going to explain it again but it better be associative the group has to contain one identity element e so in other words if we have e and we use the operation with a it’s the same thing as a operation e which just gives us a back and for every element in our group there better be an inverse so we better have that g dot g inverse is equal to g inverse dot g and this just gives us the identity element back of course if there’s no identity element we can’t have inverses so we’ll talk about what those are called in the next video so four axis closed associative identity inverse and we’re going to check all of these when we do problems about figuring out whether things are groups or not so one example of a group is that this addition modulo 3 in other words the remainder of a number if you were to divide it by 3. so if we take two and we add two together we get four we take the remainder we get one back that’s how i’m going to explain it if you want to know more about modulos check out a discrete math course uh for the sake of linguistics that explanation is close enough to how it works that i think it’s good enough okay so we have to check first is it closed and how do we do that well we make sure that every element is in z3 and i’ll tell you right now z3 is just the set 0 1 and 2. and then when we do an operation together so let’s say plus in this case on two elements a and b it better come back into that set zero one and two in fact if we take a look at the chart here no matter what we do we end up with a zero one or two back so it is closed that’s good two is it associative and i urge you to check yourself i’m not going to go through it because it’s really tedious to make sure all combinations work but yes it is indeed associative the third one which is a little more challenging to figure out is there an identity so in other words what is the identity with the plus operation equal to well we need to find an element that takes everything that’s paired with back to itself so a good place to start is zero of course because we know with addition zero is usually our identity element so if we take zero plus zero we get zero back zero plus one we get one back zero plus two we get zero two back and similarly on the other line one plus zero is one two plus zero is two so yeah zero is going to be our identity element and finally four is there an inverse for every element which means if we take zero plus zero inverse we better get zero back if we take one plus one inverse we better get zero and if we take two plus two inverse we better get zero so what are these inverses well 0 what takes us back to 0 is 0. well it’s just 1. so 0 oh sorry it’s just 0. 0 plus 0 gives us 0. what about 1 plus 1 inverse what if we have one what takes us back to zero well two takes us back to zero so the inverse of one is going to be two and if we have a two what takes us back to zero well one takes us back so every element has an inverse the inverse of zero is zero the inverse of one is two and the inverse of two is one therefore it has an inverse it has an identity it’s associative is closed so this is a group that’s all it takes to satisfy these is these four axioms in fact this group is also commutative which means that we can switch the order of things around and we can tell that it’s commutative based on kind of the symmetry here but we can also check ourselves manually and if it’s commutative then it is called billion group so abelian here just means commutative we don’t say commutative groups we say abelian groups of course contributing to the person who discovers this historical note not really important but that’s a terminology you should know as abelian before any of these things just means it’s commutative as well okay so here’s a question is the group consisting of truth values 0 and 1 and exclusive or the group so of course this is the exclusive or these are our truth values so the first thing we should do of course when it’s small enough is just draw a chart so we have the exclusive or operation we can have zero and one as our first argument we have zero and one as our second argument what is the chart well exclusive or is true when the values are not the same so it’s going to be true when we have a zero and a one it’s going to be false when the values are the same so our chart looks like this so we have to check one is it closed and if we take zero and one and we put it with an operation and get zero and one back of course it’s closed we can only get one not for available thickness of this pen uh is this equal to one exclusive or zero exclusive or zero so we can check these individually well one exclusive or zero is just one because of or zero is this equal to one exclusive or well zero and zero is going to give us zero back because they are the same value so it’ll be one inclusive or zero because one is or zero so just be one equals one so what the one example we did is associative in fact if we did all of the examples we would see that is associative so it is associative you’d have to check it manually or find one counter example so good luck uh the third is their identity so other words is there an e for exclusive or so is there something that always takes you back to its own value and there is so if we take a zero with zero we get zero back if we take zero with one we get one back so our identity element here is actually zero and this intuitively is kind of weird because you’re like well it’s just a it’s just a matching thing if the values match you get false if they’re the same you get it true but the way it functions the values it gives back shows us that this zero is an identity element so if we take zero with zero we get zero back if we take zero with one we get one back and that’s enough to satisfy the identity element because whatever it goes with on the right it gives it back okay what about inverses let’s ask ourselves if there are inverses for this so what is the inverse of zero and what is the inverse of one so what gives us back to zero well if we have the same value we get zero back so the inverse is just itself so it satisfies all four axioms therefore yes it is a group okay so groups that are commutative are called abelian groups i just went over this but some groups that you may want to verify yourself that are in fact groups are the real numbers and multiplication so we know that 3 times 2 is equal to 2 times 3. it also satisfies other group properties so for instance if you take a real number and multiply it with another real number you’ll always get a real number back another example would be the universe of sets and intersection so a intersection b is equal to b intersection a in fact i really advise you to check all of these and check if they’re groups so find the identity element find the inverses um see if it’s closed see if there’s any counter examples to associativity there shouldn’t be and finally the third example with the integers in addition again of course we have the two plus three is just equal to three plus two and it satisfies all the other group properties so again please check on your own it’s a good exercise um this is probably something i put in an assignment just to check these things because i mean you’ve worked with these before right if you’ve been watching this series and you’ve been doing stuff in class these sets are things you’ve been working with or these groups anyway okay so here’s the theorem and i’m going to do one proof because i think proofs are kind of important and i kind of want to show you the reasoning for this but it’s a very important theorem so a group has only one identity element so you might think to yourself i have a group here and it has two identity elements well if that’s the case something is wrong because there is only one identity element and this can be proven for any group so here is the proof i’m just going to let g equal g and some operation be a group because our assumption is we’re working with groups here okay and now what i’m going to do is i’m going to assume that we have some e and e prime are identity elements so i’ll just call them identities and the important thing here is that e is not equal to e prime so they’re not the same identity elements that’s i’m going to assume this and i’m going to show that if we assume this we get some terrible contradiction okay so what does this mean well this means that e dot a is equal to a dot e which is equal to a and this means that e prime dot a is equal to a dot e prime which is equal to a as well so here’s the clever part in fact to to some people this may be super clever for some people this may just be the most obvious and stupid thing you’ve seen but for the sake of showing how this proof works let’s just assume that i have an element e on its own and i’m going to start with e well what is e well if we take the identity element e prime and then use the operation with e well this is just equal to e right so this is the same because this e prime is an identity element so it should map e back to itself but what do we know about this group well identities are commutative identities will always be commutative which means that we can flip around e and e prime so we can take e dot e prime okay so if we do the same thing as before and we take this as the identity element which maps the second one back to itself we also know that e is an identity element which means that it can map this e prime back to itself which means that this is just equal to e prime so we’ve found that these two identity elements are actually equal which means if we have two identities and we assume they’re not equal to themselves eventually we come to the conclusion that well actually they are equal which means by some contradiction and reasoning here that we can only have one identity element because if we have two and we say they’re not equal it turns out they’re actually equal so any group will have exactly one identity and no more and we’ve proven this generally so this wasn’t proven for some specific group this was proven for anything so that’s it for this video if you have any questions please leave them in the comments below and i will do my best to answer them

rnrn