 okay let's talk about closure property we are interested in closure property for integers but first we need to know what is closure property so let's say there is a set we all understand what set is and there are few numbers like this and now there are different operations that I can do with the numbers and sets the operations are usually addition subtraction multiplication or division and there are other many operations which we do not really need to know for now but let's say just for these four and even for these four we will only focus on addition and subtraction here so let's say this is the only set that we have for now set a which has four numbers and what if we select any two numbers and perform these two operations okay so let's choose any two what I can choose is two and four now if I add two and four the result is six right I want to focus on the result here then I choose two more numbers say three and four again so now three plus four is seven let me choose subtraction operation now and we will choose three and five so let's say I write five minus three now let's look at all these results do we find these results in the set a six is not present in the set seven is not present in the set two is present in the set but for every subtraction would there be the result available in the set say if we chose to write three minus five instead I would get minus two and this result is again not present in the set when the results are not present in the set we say that a given operation say addition or subtraction is not closed for the given set but when the outputs are present in the given set then we say the closure property is satisfied for the given operation now this set was small the sets that we usually deal with are larger sets such as set of whole numbers now whole numbers are natural numbers and zero so the whole numbers could be represented by w and I'll write 0 comma 1 comma 2 comma 3 comma 4 and so on what we are particularly interested in this video is closure property for addition and subtraction for integers so we will have negative integers then minus 3 minus 2 comma minus 1 this is how I usually write this and this is how we can represent the set of integers now let's talk about closure property for addition right and subtraction for these two sets so what we're going to do is we are going to select any two numbers and we will perform addition and subtraction and we will see whether output exists in the same set or not now let's choose integers as a set first so just think about any two numbers I can think about say minus 4 and minus 1 if I select minus 4 and minus 1 but I'm adding those so I get minus 5 is minus 5 inside i is minus 5 an integer yes and you can you can verify this for any two given numbers but it will take so long to verify all the combination of all the numbers and check the addition of it and that's why we have this property on platter that integers follow the closure property for addition and so we know that if you select any two given numbers you will get an output as an integer an elegant way to write this is that integer set or we could just write integers is closed under addition as well as subtraction so even if you subtract any two given numbers from the set of integers you will get an integer for example if you select 2 and minus 2 so 2 minus minus 2 gives you 4 which is there in the integer set so this is a very important property that we just learned about integers that it is closed under addition and subtraction what about whole number set let's see whether whole numbers are closed under addition and subtraction select any two given numbers from the whole number set say 1 and 2 and if we add them do we get whole number yes is there anything that doesn't follow this property can you find the combination of two numbers such that the output is outside of the whole numbers no so whole number set is closed under addition but I haven't written subtraction here and the reason is that the whole number set is not closed under subtraction let's see why there are certain whole numbers when you select them and subtract them you get negative numbers and in whole numbers you don't have negative numbers for example if you chose to subtract 2 from 1 so if I write 1 minus 2 this is a whole number and this is a whole number and you will get minus 1 and this is not a whole number and so because of at least one case where the output of subtraction of two whole numbers is not a whole number it makes the whole number set not closed under subtraction so that is also very important so we can write whole number set is closed under addition but not closed under subtraction