Bringing the set operations together

let's now use our understanding of some of the operations on sets to get some blood flowing to our brains so I have to find some sets here and just to make things interesting I haven't only put numbers in these sets I've even put some colors and some little yellow stars here and what I want you to figure out is what would this set be this crazy thing that it involves relative complements intersections unions absolute complements so I encourage you to pause it and try to figure out what this set would be well let's give it a shot and the key here is to really break it down work on the parentheses the stuff in the parentheses first just as you would do if you were trying to parse a traditional mathematical statement and then it should hopefully make a little bit of sense so a good place to start might be to try to figure out what is the relative complement of C in B or another way of thinking about this what is B minus C what is B if you take out all the stuff with C in it so let me write this down the relative complement of C and B or you could call this B minus C this is all the stuff and B with all the stuff and C taken out of it so let's think about what this would be B has a zero does C have a zero no so we don't have to take out the zero B has a 17-2 C have a 17 yes it does so we take out the 17 B has a three but C has a three so we take that out B has a blue C does not have a blue so we leave the blue in so let me write down we leave the blue in and then B has a gold star C also has a gold star so we take the gold star out so the relative complement of C and B is just the set of zero and this blue written in blue so let me write this down let me write that down now it gets interesting we're going to take the absolute complement the absolute complement of that so let me write this down so B be the relative the the absolute complement of this business is going to be all things all let me write it this is the set of all things in our universe set of all things in universe in universe that are neither that are neither a zero or a and I'll write it in blue or a blue that's the only way I can describe it right now I haven't really defined the universe well we already see that our universe definitely contains some integers it contains colors it contains some stars so this is all I can really say this is set of all things in the universe that are neither a zero or a blue so fair enough so we so far we figured out all of this stuff all of this stuff let me box this off so that is that right over there and now we want to find the intersection we need to find the intersection of a and this business the intersection of a and that business let me write that down so it's going to be a intersected with B the relative complement of C and B and the absolute complement of that so this is going to be the intersection of the set a and the set of all things in the universe that are neither a 0 or a blue so it's essentially the things that satisfy both of these that has to be in set a and it has to be in the set of all things in the universe that are neither a zero or a blue so let's think about what this is so the number three is in set a and it's in the set of all things in the universe that are neither or zero or blue so let's throw a three in there the number seven it's an A and it's in the set of all things in the universe that are neither a zero or a blue so let's put a 7 there negative 5 also meets that constraint zero does not meet that constraint a zero is an A but it's not in the set of all things in the universe that are neither a zero or blue because it is a zero so we're not going to throw a zero in there and then a thirteen is in a and it's in the set of all things in the universe that are neither a zero or a blue so we could throw a 13 in there so we've simplified things a good bit this whole crazy business all of this crazy business has simplified to this set right over here now we want to find the relative complement of this business in a so let me pick another color here so we want to find the relative complement of the business in a so now I'll just write you write out the set three seven negative 5/13 actually let me write out both of them just to make it just so that we can really just kind of visualize them both right over here so a a is this it is three seven negative 5 0 and 13 and I could write the relative complement sign or actually let me just write well let me write relative complement I was going to write minus and so in all of this business we already figured out is a 3 a 7 negative 5 and a 13 so it's essentially start with this set and take out all the stuff that are in this set so this is going to be equal to so you see you have a you're gonna we're going to have to take out a 3 out of this set we're gonna take out a 7 we're going to take out a negative 5 and we're take out a 13 so we're just left with the set that contains a 0 so all of this business right over here has simplified to a set that only contains 0 now let's think about what B intersect C is B intersect C these are all the things that are in both B and C so this is going to be B intersect C let's see we don't 0 is not in both of them 17 is in both of them so we'll throw a 17 there the number 3 is in both of them the number 3 is in both of them blue is not in both of them the star is in both of them so I'll put the little gold star right over there and so that's B intersect C and so we're essentially going to take the union of this crazy thing which ended up just being a set with a zero in it we're taking the union of that and B intersect C and we deserve a drum roll now this is all going to be equal to we're just going to combine these two sets it's going to be the set with a zero a 17 a 3 and our gold star and our gold star and we are we are looking at I should make the brackets in a different color and we are done