O Level Math / High School – Sets

hello in this session I’m going to talk about sets of mainly be going through the basic ideas to get you started with a good understanding of the kind of meanings and notations that we normally use The Awl level mathematics a set is basically a collection of objects it’s it can be any object that you can imagine it can be a collection of numbers a collection of letters or collection of fruits and chairs and tables and books and pencils and anything you like so for example it can be 2 5 & 7 that’s a set it can be the letters a G that’s another set it can be a cup and a sausage that’s a scent as well just a collection of anything is really quite general right well we do at all level get questions with sets that our numbers Corrigan but we all we do also get questions about other objects sometimes right now when we talk about a set when we pull this start trying to describe it more carefully we need to learn some names now if I have a set of objects each objects in this hand is usually called and elements right so an object in this set for example is called and cannibals so instead of calling it an object when you use the word elements in mathematics we understand that you are talking about the thing inside of sets or the the member of a set and they would often represent then a few of these elements belong within a set by using curly brackets like that so this are called curly brackets so you can call them braces let me write down these words called curly brackets or can fully embrace so there we have three sets okay and having written down particular said we might want to talk about it sometimes we might want to refer to it it would be quite troublesome if I go around saying my set of numbers to five and seven or my set of things Apple Cup in sausage that’s quite a mouthful you have to spell everything in your sense so what we normally do is we give each set the name and again in and call it anything

you like in mathematics we tend to make it simple by giving it a capital letter right so for example I might call this an eight so the name of the full Sun is a so next time I tell you I mentioned a set a you know that I’m talking about the set containing two five and seven so that makes it a lot easier I don’t have to go around saying the set up to five seven all the time the next one I might call it e the last one might call it and so the set if I tell you what that I’m talking about the set M you know what I mean I mean the set of Apple Cup and sausage so I don’t have to spell every word deal so that’s naming of a sets this capital F Thursday I just but now another thing that we would like to do about the sand is that we would like to know or tell how many animals there are innocent now if I write out the whole set then it’s quite obvious how many elements there are in the set a I’ve three elements in the set B I have one two three four elements right no but if I just tell you the set a without telling you what are the the elements okay if I don’t tell you if I’ve not told you about those elements yet then you would not know how many elements there are in a so in mathematics we have notation to represent this number of elements it’s written s it’s written s and if I’m interested to know the number of elements in a a inside so if I see n bracket baby this means the number of elements okay of course the just writing now this symbol does not tell me what the ribbon is it simply means that this symbol represents that number so for example if in this case I know that the most and with three I was right and bracket ne is equal to three so what we are learning here is the meaning of this symbol litter and with the brackets so we it’s a very very special case you can imagine having a scent that has nothing inside so it’s a bit like you might carry a bag around with some objects inside and that is it’s like a set but you might also carry a bag around then has nothing an empty bag so an empty set right it can be fight simply represented as using the two curly brackets with nothing inside so it’s obvious that isn’t that nonsense now another way to represent it is by using I think zero and drawing a line across it so that symbol also represents an empty sense right so just a set with nothing inside we will go on learning a few more new patients the symbols that we can use

later on and we answer questions now there is this symbol to me is the symbol is like that okay sort of like a you rotated by 90 degrees and before my coming through the center so this symbol means is an element of how do we use this let’s look at an example right if I have say Santa said a again maybe not again but just maybe I’ll just make up something like two seven eight okay I said a elements to seven and eight so for example I know that seven is an element of a so I can of course just say that in words but another way to to show this idea is to use bats and all that I can say right – followed by that symbol followed by a team of seven okay so this this symbol here means that seven is an element of a so that’s what that symbol means okay so rather than saying out the whole sentence this makes things a lot simpler to write but of course the disadvantage is that you have to remember what that symbol means likewise we have another animal which means peace now you start with the same elements symbol as before but we strum on line across it so this means is not sin and even self so using that example again this side I only have two elements 2 7 & 8 so 3 for example is not an animal from 8 right 3 so this means 3 is not send any ones of a right the Knicks anyone the next symbol is a subset of looks like this so like the previous symbol you rotate the 90 degrees but without the line in the middle now subset means what it usually means a smaller set so for example if I look at that example again hey I said it was to 7/8 now if I imagine making up a smaller set using say 2 & 7 2 & 7 both with all to 8 but it’s a smaller set so I can call it I can use this symbol this symbol here and write a thanks to it and what this means is that this Sun

containing 2 & 7 it’s a subset of a all right that means that the elements in this set are also contained in a now but the although that’s usually often the place the street meaning of this symbol this the meaning of this word subset it can it just means a set whose elements contained inside a so for example a subset can actually be the whole set okay so if so so a itself is a subset now these are special cases a itself it’s actually a subset of a alright that’s because of the the way the meaning of the word subset is defined it just means as long as I’m an animal as long as I’ve said whose elements are all inside a then that cell is called a subset so in the special case when when the Sun is a itself the animals of a are indeed all inside a so therefore we can also say that a is a subset of a it’s a special case does not mean that PHA is smaller than a necessarily now another special case is the empty sense the empty set is a subset of a now that might sound rather odd an empty set well then he says nothing inside so so as I saying nothing is part of something but if you think about it that kind of makes sense because if you have nothing then he it can be part of something right so in the sense that if there were several things and you imagine including nothing inside it’s basically the same thing okay I shall try to explain further of this but just take note that we have these two special cases the empty set and the set itself both subsets of of a the next symbol is equal to now that’s very familiar we have learned all about this in arithmetic and in algebra now but in what was maybe use it for sense we have to be slightly careful so let me use my old and some for the game a three numbers say two five right I have this set here now if I have another set P which contains the same element to five and seven then I can use this symbol to say that V is equal to eight that’s clear enough they’ll be exactly the same but what if I have set C whose elements are five to seven is C is the set C still the same as a you see a is to five and seven six five two and seven the ordering is different well in the

mathematics of set we say that they are still the same because a set is only defined or described or specified determined by what animals it has right is not described by the ordering of the so it does not better how the elements are I’ve reached if you just imagine if you have a bag full of things all right it does not matter how the things I have been she can be in the complete man so it could be arranging it roll some order so as long as the elements are the same it does not matter how they are ordered so therefore C is also equally so that’s the important thing to know if ordering is not important so that makes the next symbol is a curly e so that’s actually an e now this symbol is called a universal set right you look at some examples in movement but broadly speaking Universal set is the largest sense that that is it usually means the largest segment we are talking about something so for example if you are talking about students in the school who play football or who plays music so the largest set when we talk about the students will be all of the students in that school for example okay and when we talk about numbers and this 1/2 or decimal numbers negative numbers and so on so this would be the set of real numbers right the set of real numbers refers to all the possible numbers in a positive negative fractions and decimals and whole numbers and integers and so on so these are universal sets of course we can always think of as sandwiches bigger than the universal stand and they’re talking about but the universal set is not be it has to be defined specified for particular problem and it’s usually and you usually just think of the biggest possible set that you’re interested in so let’s look at some examples in a moment next we look at the problem of how to describe a sound well actually we’re going to look at the notations we need to use to describe the sense right we can always describe the set verbally we can try for example let’s think of all the numbers that are between 1 and 10 right so I can describe this in words of course now but if I describe in words it tends to be when gone and in mathematics we often like to have notations to make it shorter and simpler to describe so in I could use algebra for example to represent a number X so that’s my number

and if this has to be between 1 and 10 it means that it has to be more than 1 and 10 and if I’m interested to say that this collection of of all these numbers in between more than 10 belong to a set that a set contains these numbers then I would write it this way I would put see I will put an X before that with Coburn all right so it just means X is the element of this set column means that X obeys that condition and the condition is that X must be between more than ten and the show that is a set of curly brackets around the whole lot so when you see this funny notation right within the curly brackets you have an axe and you have a colon then whatever comes up to the colon is the condition that tells you the room rate that X must obey so in this case what it means is that as long as the number X is between 1 and 10 then all of these possible values of X belong to this set so this is a well say this is a shorter way than the way of writing describing this set compared to breaking the holding or in words but of course if you’re seeing this for the first time you can look really confusing but we will get used to it with some examples so you right so that’s one example so let’s say following on from from this I’m I’m looking at X that is between 1 and 10 again no but this time I am only interested in X that there are positive whole numbers right so now the positive whole numbers is let me represent this set of positive whole numbers by the letter n so that’s at this set and first so actually positive whole numbers another name for the set of positive whole numbers is natural numbers okay that’s why I use the capital letter and to represent this set now suppose that have interesting in a set in which the admins ax is a positive whole number and it is also between x equals 2 it’s also between 1 and 10 now the way that I would describe the set right where X is a possible number is to say X is an element and if it is also between 1 and 10 I would then write : up to that and then one lesson next less than 10 I do sure that this is a set i put curly brackets around them so now i have again a short compact notation that tells me that describe this sense right rather than saying that X is a positive whole

numbers and it is between 1 and 10 so that’s a very long sentence I write it down this way all right using the symbol this symbol this symbol and those symbols and if I read this out in English what it says is that well for me to curly braces it tells you that it’s a set so this is a set of X which is an elements of the set of positive whole numbers and you know based condition that axis more than one and less than ten meaning then it is between 1 and 10 so notice that to describe the whole thing in words I mean to say many words is a pretty long sentence so this way of doing it using using the symbols is quite short it gives us a short and quick way of writing this down and the main thing to to learn here is to understand the meaning of each symbol and understand what it means when we put the whole lot together let’s be able to look at Venn diagrams then diagram is a way to to represent the sense to visualize the Sun using pictures so that’s the easiest ways to look at some examples and some pictures right just a to understand so noticed I was doing yes I’m defining my universal sense so this is something that that we do it is is the largest sad that I’m going to use all right but that was that I’m going to talk about things I’m just going to make up or look and sense that are smaller than this and and that contain these animals here so that so I not want to talk about us in this particular discussion right so if you like that’s how we we can we can think about Universal set you just make it up you just or you just decide on on the biggest Sam order because collection of objects that you think you are interested in and then you just discuss it or think about it or calculate with it so right that’s my universal sense and I’m going to define some sense inside this Universal set so with the defined set a okay beside a of things anyone’s two

four six eight okay so to sense here now I want to represent this using a picture so what I could do is draw a rectangle and say that this rectangle represents the universal sense okay now since a have elements that are inside the sets and a is also smaller set I can represent a with a shape inside the rectangle okay so me represent the idea that a is a smaller part of the set e of the universal set by drawing a shape within a bigger shape Victoria we have doing it so that’s my hate okay and I can just to make it quite clear what this mean I can write out these are the ones in the picture so a is two four six and eight I can write four four six and eight inside the circle and outside a still within the rectangle I can write out the rest of the elements which would be one three five and seven it does not really matter where on the picture I write these numbers as long as they are within the rights region all right so the the elements of a should be inside the shape of a the elements that are not in a should be outside the ship of me but everything should be within the biggest rectangle the the big rectangle which represents the universal sense so everything is inside the universal set when understanding things that are outside so that’s a Venn diagram write the name given to such a picture is right no let me think of another sense I call it the sand B which is equal to two and four plus you can see that B is a subset of pain right it’s more than a and both elements are in pain now if I want to show that on the Venn diagram well I could – I noticed I already have the two and for that I could just draw a shape around it he lives and that’s my B it’s about I have a Venn diagram for this picture for the subset of a just a smaller shape within the circle I’m going to make up another set C which is form I will contain the anyone’s 6/8 right so now I want to draw a Venn diagram fault a and foresee it was a set e to set a now it could be careful here what is that a and C had that it was six a in common alright but the only one five seven and two for are not in common so when I draw

the shape of a and C I must remember that there some elements that are the same but so many months that are different the way to draw such a big choice to draw two overlapping shapes also okay and in a place where they overlap we write down the elements that are common six and eight okay and outside of that overlapping region we have five and seven see there is the five and seven are inside C but not but outside a whereas a would have two and for 24 I don’t say a but outside C so this is how we would represent to sense that some but not all of the animals in common okay and now and you notice that there is this overlapping widget here now there is a symbol for that it is for intersection the symbol is an inverted u right so in this case I would write this overlap in region and the two animals that using a symbol for intersection C so this means a intersect C and it refers to the common element six and eight two six and eight here okay so any in the same C is the set of elements that are present in both a and C so that’s intersection now we can have another simple now Union refers to all of the animals in the two cents so if I’m right a union C okay then we’ll include all the animals so it would include two four six eight five seven that’s Union now go to burn one more having to learn one more symbol that’s in boys court compliments the symbol is an apostrophe so that’s rather odd symbol now let’s see how you would write this let’s look at the animal a okay if I write down a apostrophe it means a meaning compliment it means all the animals that are not in a so a contains these animals ‘ contains the rest the rest that are stealing the universal sense so this would be you know it would not be two four six and eight so it would be one three five and seven a compliments and where would this be

now if I draw the circle to represent a which contains all those animals then a compliment refers to the region outside the circle so we can imagine that the area inside the circle represents a and the area outside the circle represents a compliment right so because because of this idea about the area representing the set it is often convenient to shade the regions that were interested in so for example if I’m interested in in emphasizing or showing this particular set a compliments and my doing by shading the region that represents a compliment like that so in the exact questions on set you know humans sometimes see a picture with some shades on it and we’ve parts of the area shaded so this is what it means all right it means that it’s highlighting a set that corresponds to that area so in this case the shaded area here will correspond to a complements meaning does the scent that is the set of animals that is outside painting that does not include the animals okay right so let’s finish off by looking at some examples about this shading idea let’s look at this picture the question is to represent the shape that the set for the shaded area using set notation z’ now what we do them in set notation set notation means what we have just seen like intersection the symbol for union the symbol for complements the symbol for okay we don’t need subsets and animals here right so it would be maybe complement Union and symbol and sorry Union an intersection and the names e a and B so let’s write down the thing that we need to use we need to use symbols e le B intersection Union and the apostrophe the continents so using these notations yeah we have to make up something string them together

somehow to to give something that is equal to a set corresponding to the shaded region right so let’s try and do that now that’s a rather odd looking shaded area but what we could do is we could look at the unshaded area now n is not treated right B is not shaded now the whole of this unshaded area is actually actually contains all the animals in a and B now if you remember that means Union so any Union B is actually the unshaded area okay but that’s not what we interested in we are interested in the shaded area which is everything outside hey you need me now but we know what we have just done what that is that is the complement so everything else are Union bees but we need to the bracket around it to make sure that we include everything a union B bracket complements so this describes the shaded area so that’s how we do this particular question right that’s the shaded area now so again a brother another rather odd or this shaped region looks like a moon right so how are we going to make up a simple some rotation to describe this sense so we think of this region as right now let’s look at whole circle here the whole circle is the same B now but in this case the shaded area is is less than B is less pie by this bit here okay so how can we represent that right let’s try this the shaded area must be inside be right the shaded area must be outside babe let me write this now it must be inside B must be outside I know how site a is given by that symbol the complements so a complement is outside of a okay so a comfortable means everything that is outside a B means everything that is inside this this P circle so it must be both the shaded region is in within B is in the set B and it is also in that set a compliment meaning that is outside of the a so because the shaded region must be in that side and that sin is common to both sets we use intersection to represent that because that’s one intersection intersection means the elements that are common to two sets and that’s the answer the Union a all sorry B intersect a complements this okay we will stop here for this session see you next time