Section 2 point 3 is on Venn diagrams and set operations so just like in algebra there were four main operations addition subtraction multiplication and division sets also have operations but their operations are a little bit different than the ones we would see in an algebra course before we get into those operations though a universal set which is symbolized by a capital u is a set that contains all the elements being considered in a problem so there are a couple different ways that they can present you with the universal set if it’s a Venn diagram you’re gonna see a u in the upper left hand corner of that box of that Venn diagram and that’s their way of saying hey every element you see in this box is a potential answer it’s part of that Universal set you’ll also see later in this video when it says u equals that’s their way of labeling the universal set here are all the possible answers you could have by the end of this question is really what they’re saying so looking at example 1 we’re gonna use the Venn diagram to determine each of the following sets so we are going to list all the elements in the following sets please be aware by the way if you see Roman numerals in these Venn diagrams these Roman numerals are labels for certain regions in the diagram the Roman numerals are not elements so when they’re asking you to type elements please ignore the Roman numerals so I’m going to put a slash through them since they want elements we’re gonna ignore the Roman numerals for this question so the actual elements are a little abcdefg all right so starting with question a where what is set a they want us to determine set a so visually that of course is everything inside circle a right here so what is inside circle a well it’s a set we’re still talking set so yes we do unfortunately need to keep drawing those curvy brackets and inside circle a in any order I see a little a I see a little B I see a little C and I also see little D close the brackets don’t hit spacebar you would just type it in just like that using the keyboard to get in those curly brackets and the commas and that’s the final answer for set a or circle a now looking at B so Part B says the set of elements that are in B but not a so I don’t want a it says but not a so let’s get rid of a so now what’s left in circle B and then I get rid of a well the only thing left in circle B is that little e so little e inside those set brackets and that’s the final answer the set of E all right now looking at question C the set of elements in the universal set that’s what that capital u means that are not in a so again we don’t want a so I’m gonna X out a so everything left in the universal box well we still have a little e that’s not encircle a and then we got little F and G are not encircle a but still in the universal box as well so that’s my final answer e FG notice the elements when the elements are letters they need to be lowercase because capital letters did the sets themselves so little letters or elements looking at D no D says the set of elements that are in the universal set that are not in a or b so I don’t want a or b it’s the only thing left in the universal box if I get rid of the circles are little F and little G I really just want to list the elements ignore the Roman numerals the Roman

numerals are labels not elements and then finally I didn’t have enough room to put it on the bottom so I put it over here they want you to type in the universal set that’s all the elements that are in the box in any order so that’s a through g a b c d e f g is the universal set those are all the possible answers that you could have the final answer the complement of set a which is symbolized by an apostrophe so when you see that apostrophe attached to any set that is going to be read out loud as complement so i’d read this out loud as a compliment it’s a complement of set a symbolized by a complement a apostrophe is the set of all elements in the universal set that are not in a so it’s not as bad as it sounds and that this definition makes it sound a little bit harder this idea can be expressed in set-builder notation as follows so we see that it is in the universal set however it’s not in the set itself so my memory trick whenever I see that apostrophe I’ll put apostrophe equals it means what’s not in that set that it’s attached to because it doesn’t have to all only be attached to a like in the definition here so it’s not in the set that the apostrophe is attached to but still in the universal set as you remember the universal set are all the possible answers you can have by the end of the question but still in the universal set you might want to even jot that down especially when you’re preparing for the test on this chapter some sort of memory trick for each of these set operations can definitely help another memory trick I’ve heard for this one is it’s the opposite of that set so what are the opposite elements so looking at example two once we see it in action we see that it’s a little bit easier than it sounds so looking at example two it says to find each of the following sets u equals remember whenever it says u equals that means Universal set in other words all the possible answers we can have by the end of this question so the universal set or all the possible answers are the numbers one through seven they give us also two subsets of the universal set they have subset a which is the set with one three five seven in it and subset B which is the set with one two three in it so using those sets question a wants us to find a complement so I do see that apostrophe and remember putting it together using our guide on the side here remember whenever you see that apostrophe it means was actually not in that set but still in the universal set so because the apostrophe is attached to a really this means what’s not in set a but still in the universal set let’s put a little curse a capital curse of you for universal set and when you do this set operation this is our first set operation there are three in this video the first is complement so whenever you do a set operation your answer is another set so we do still need to keep drawing those curvy brackets so I drew my curvy at my first curvy bracket so it’s not an a well if I look above a has one three five and seven in it so you want what’s not an A or in other words the opposite of said a but still in the universal set so if I get rid of one three five and seven from the universal set what’s not an A but what’s left over in the universal set would be two four and six so those are all the numbers that were not in a

but we’re still in the universal set you always always look at the universal set whenever you see that apostrophe that compliment symbol and it doesn’t hurt either to check your answer to make sure you didn’t forget any of them so if a was the set with one three five seven the opposite of a would be two four six so you should never ever have an element that is both in a and its complement because complement means opposite of so they should have all the opposite elements and its complement this is the set with two four six would be the final answer let’s try another one if we look at B once B complement because I see that little apostrophe attached to B so putting it together this actually means what’s not in set B or the opposite of set B but still in the universal set of course still in that universal you set so B complement well if I look at B now B has one two three in it so comparing to the Universal set if I get rid of one two three all the leftover elements in the universal set that we’re not in B are four five six and seven so if set B was one two three its complement is four five six and seven as the final answer the next set operation is called intersection the intersection of sets a and B which is written with an upside-down U shape or like a horseshoe shape opening down so whenever you see that U shape opening down in between two capital set letters that is now going to signify intersection we have a lot of use in this section we have the universal u we got the intersection U and then we’re gonna see on the next slide we have what’s called a union you but whenever that U is opening down that is intersection U and that intersection is the set of elements that are common to both sets a and set B this definition can be expressed in set-builder notation as follows so in their notation they have the set of X such that X is an element of a and X is an element of B so the key words were what’s common to both sets now in the definition they use sets a and B but it doesn’t always have to be those particular letters so the idea is what’s common to both sets so when I see that you shape opening down when I see the intersection symbol my memory tray is I think matches where do these two sets match at that’s the intersection looking at some examples example three wants us to find each of the following intersections they have u equals remember when it’s u equals that means the universal set is the universal set or all the possible answers by the end of the question are the numbers 1 through 7 the universal set equals the set of 1 through 7 subset a is the set with 1 3 5 7 sub set B is the set with 1 2 3 and sub set C is the set with 2 3 4 5 6 so they give us a bunch of subsets of the Universal set to work with so we’re gonna use these sets for Parts A B and C so a says a intersected with B so if I bring me sets down a was 1 3 5 & 7 and I want to intersect it with set B which was the set with 1 2 & 3 and you don’t have to show all this work I’m just trying to write everything out so I can explain it better but if you want to go right to the final answer when you’re working on your my math lab assignments that’s fine but again as soon as I see that intersection symbol my memory trick is matches so where do these two sets

match well they both have a 1 and a 3 in common and whenever we use a set operation like intersection our answer is gonna be another set so we’re still gonna have those curvy brackets but we do have the set with 1 3 as the final answer I’m gonna erase it so I can move this answer up so that I have more room to write with the other parts here we got a final answer of the set with 1 & 3 looking at Part B now B wants us to do a intersected with empty set so a again was one three five and seven and I’m gonna intersect that set a with the empty set which is no solution it has no elements in it whatsoever so again I see the intersection symbol I immediately think matches so where do these two sets match well they have nothing in common they have no matches so the answer is gonna be no solution because there are no matches so I get empty set again as the final answer keep in mind the homework might have the empty set written this way or the other way was set brackets that have nothing but Arunima they were literally empty so either way we still get empty set as the final answer whenever you intersect empty set with any other set in the world and then finally looking at Part C C says a intersected with the U u meaning the universal set so a again is still one three five and seven and we’re gonna intersect it with universal U which was the numbers one through seven and it looks like I’m running out of room so I’m gonna move my answer to Part B up to create some more room here so he said B was the empty set again there we go and now as I look at a and the universal set where do they match will they match it 1 3 5 & 7 so that’s gonna be the final answer now you could leave your answer like this or if it’s a malt choice question in the MyMathLab homework you might not see this listed as a potential answer because what happened is is we got one of the original sets back again we got set Eddie back again so the most likely would just abbreviate it as set a so you could say well we got the set with one three five seven back again as the final answer or you could just simply select capital a for set a we got set a back again when we intersected it with the universal set because those are all the matches so set a final answer to Part C the third set operation we’re going to work with is Union in this course so we’ve dealt with compliment intersection and now Union so the union of sets a and B which is written by surprise surprise another you so this use opening up you might be saying well how do I know if it’s a union U versus a universal you well you’ll just have to look at the context of the question if that you that’s opening up is in between to set letters to capital letters then they’re referring to Union u whereas if the U is by itself or if it’s like u equals that’s the universal u meaning all the possible answers you could have by the end of the question so we have two different use here so back to Union u so the union of sets a and B which is written a union B or we could even say a United with B is the set of elements that are members of set a or set B or both sets so here’s their fancy set builder notation where X is an element of a keyword here’s or or X is an element of B so what does that really mean well it means we don’t care about matches anymore like we did with intersection my keyword when

see that Union you is just to simply combine the sets together or list them all out who cares if the sets match just list all the elements you see out so you’re gonna combine the sets because of the word or meaning we’re happy with all the answers in one set or the other set or both just merge them all together combine them all together and that’s all you do with Union so trying it looking at example for the universal set is gonna be the numbers 1 through 7 again set a is 1 3 5 7 again set these 1 2 3 set C is the set with 2 3 4 5 6 and we’re gonna find each of the following sets so starting with a this is a union B or a United with B so if I bring down the sets to see it better set a was 1 3 5 and 7 and I’m gonna do a Union you I’m gonna unite it with set B which was 1 2 3 and we just look up our memory trick it might help to jot these memory tricks for the set operations in your notes somewhere so you can easily see it when you’re doing your homework but as soon as I see that that’s a union you I think combine the sets together so yes they do have a 1 in common but who cares so you just merge the sets together so there’s a 1 a 2 I’m checking off the elements as I go to make sure I don’t miss any there’s a 3 there’s a 5 and there’s a 7 when I combine the sets together if they match you only list it the element once I don’t repeat any of the elements so here’s the final answer when I merge those sets together I get 1 2 3 5 7 that set as my final answer again it’s the set operation so I do need this curve set brackets around the final answer and all of these are going to be in roster method unless they specify otherwise in the homework so I just put those commas in between all the elements so you just combine the elements together and you are done let’s look at B so B says B Union C or B United with C so B B was 1 2 3 and now we’re gonna unite it or do a union with C which was 2 3 4 5 6 and as soon as I see that Union you in the middle of those curvy set brackets I immediately think combine the sets together so we do tend to get longer answers when we do a union so when I combine the sets together I have a 1 over here I have a 2 you have a 3 again we have some matches but who cares we don’t care about matches with Union you have a 4 check we have a 5 and we have a 6 so between the 2 sets when I put them together we have 1 2 3 4 5 6 in set brackets as the final answer so combine the sets or list them all out are usually the memory tricks I hear with Union you I actually have one more example I couldn’t quite squeeze it onto this slide so let me try to erase so that I can try to find room somewhere for it here we go so I’m just gonna put it somewhere on the right side of this slide so the other example that I have here was Part C and it was set C United with the empty set so set C was 2 through 6 and we’re gonna union or unite it with empty set now don’t overthink it but try to be tricky like this in the homework where they put in the empty set just remember Union you as soon as you see it think combine the sets together the problem is that empty set has no elements in it whatsoever so empty set contributes nothing when it’s a union you so when I combine the sets together

I still get back 2 3 4 5 and 6 and empty said I might as well ignore it with Union new which was totally different than when I intersected with empty set then it dominated the answer but with a union empty set you can ignore it when you combine the sets together you get back the other set which happened to be set C so we get back set C or 2 3 4 5 6 back again for the final answer to this extra Part C that I just threw on this slide here so how can they make these set operations a little bit trickier well they’re gonna mix a bunch of operations all together within the same question so it’s important that we keep in mind what each of the 3 set operations means to do our memory tricks but also we have to be aware of order of operations yes there’s an order of operations even when we’re talking about sets in math now you might remember from algebra there was an order of operations or PEMDAS please excuse my dear Aunt Sally however you might have learned it and the order of operations are kind of similar four sets is the good news so the peon PEMDAS stood for parentheses and is the same with sets really with any type of math you’re studying you’re always gonna do inside parentheses first so the P still stands for parentheses now when we say parentheses we really mean any sort of grouping symbol it could be normal parentheses the open parentheses it could be solid brackets it could be the squiggly set brackets but if you see any sort of grouping symbol in the original question we’re gonna do inside parentheses from left to right to begin and if you see parentheses inside parentheses we will do the innermost parentheses first or work away from the inside out I could see my math lab trying to pull something like that is why I mention it so inside parentheses first after we take care of everything in parentheses then we’re gonna do the complement next that was the apostrophe symbol so we’re the apostrophe symbol from left to right next after we take care of all the complements then everything else will be last so any kind of Union symbols or intersection symbols we haven’t done yet will take care of those from left to right we always go as we’re doing these order of operations so example 5 given that you equals so that’s the universal set when it says you equals the universal set is the set with little a b c d e in it set a has little bc in it and set b has little BCE in it we want to find each of the following sets so Part A in parentheses they have a union B remember the U in between two capital letters when is opening up like that is a union you have a union be reunited with B and then outside the parenthesis I know it’s kind of hard to see on the slide here it’ll be easier to see in the MyMathLab but there is an apostrophe or compliment floating outside those parentheses so we’re going to use the order of operations so step one we said if you see parentheses you always do inside parentheses first well inside these parentheses is a United with B so we got to calculate that first so a was little B little C and I’m gonna unite it with set B which is little BCE think back to our memory trick for a Union Union means just simply combine the two sets together and between the two sets we have a little B we have a little C and we have a little e we don’t care about matches with Union just simply merge the sets together so we just figured out the answer to what’s inside parentheses is BCE now step two

now that we finished up the parentheses we look for complement that’s the apostrophe symbol and there is a compliment outside the parentheses which means you’re gonna apply the complement to the answer from step one you’re applying the complement to a union B well we said a union B if I bring it down was b c e so when I apply the complement to that set to that a united with be set think about what that apostrophe that complement means it means this was the tricky one was actually not in that setre the opposite of that set it means what’s not in that set but still in the universal set so what’s not in that set but still in the universal set or the opposite of that set so if I go back to the universal set that was the set with all the possible answers that’s the you equal set I’m circling it if we don’t want the set with BCE let’s get rid of it b c e so what’s still in the universal set after i get rid of little B little C little E is a and D so the opposite of BCE would be the letters a D and we put set brackets around it whenever we do any operation we still have those set those curvy set brackets and as I go back to the original question we’ve taken care of the parentheses in step one we just took care of that apostrophe the complement in step two there’s nothing left to do so that must be the final answer so the only thing you would type into the answer box in MyMathLab is what we’ve done last at the end of the last step so the final answer is the set with little a little D in it I’m gonna go ahead and move this answer up and then we can work on Part B so you might want to pause it to pause the video if you’re still writing I’m gonna race Theresa Reis and we said the final answer was elements ad inside those set brackets using the same sets let’s look take a look at question B it says a compliment because if they apostrophe intersected with B complement alright well we see lots of upper set operations there step 1 you would do parentheses first but there aren’t any so now I go to step 2 says we do complements from left to right the apostrophes from left to right well we’ve got a couple compliments here so as I go from left to right the first compliment that we see is a compliment and remembering what compliment means is we just mentioned a few minutes ago compliment actually means what’s not in that set that the apostrophe is attached to so when I put it all together it means what’s actually not in set a but still in the universal set it’s always compared to the universal set so it’s not in set a but still in the universal set so when I go up here I see that set a has little B and little C in it so if I get rid of little B and little C what’s not in that set or the opposite of that set is gonna be a D and E a little a little D and little e it’s the opposite of a or what’s not an A you can think of that complement is meaning is a d E and it doesn’t hurt to check just to make sure it makes sense because complement is the easiest operation I think to miss so said a was B and C

what’s the opposite of that set would be a de all the other letters and the universal set good alright so we took care of a complement done we are still on the complement step step two of the order of operations because we have another complement to now take care of so then compliment as we go from left to right is B compliment or B ‘ and we said compliment means what’s not in that set so in this case not in set B but still in the universal set so they go back up to the original questions you’re still using all the same sets from example five here set B has little B C and E in it so when I go back over to the universal set when I get rid of a little B C and E the other letters are a and D and let’s go ahead and erase all these extra markings so you can see this better just to check our answer so if set B was BCE what’s not in that set the opposite would be letters A and D good so that looks good all right so now we go back after each step we constantly go back to the original question to see what operations are left so we did a complement we did B complement now we’re on step three everything else the only thing left is an intersection symbol in the middle there of those complements we’re going to intersect our answers from steps one and two it’s for intersecting a complement with B complement and our memory trick for intersection was matches or what do these two sets have in common well they both have an A and they both have a D in common and we have those set brackets around our answer still so it’s the set with little a little D we’ve taken care of all the operations from the original question so this must be the final answer so we get the same answer as we’d got for question Part A which is the set with a D so we saw full of examples of what to do when there are multiple set operations and we have to abide by the order of operations and we keep in mind the memory tricks for each of the three operations but that was when we were given the sets what if we’re not given the sets but instead we’re given a picture of a Venn diagram like with example six here so that we can still use the same memory tricks but the thought process is just a little bit different when we change to a visual perspective so keep in mind the universal you will be always in the upper left hand corner of these Venn diagrams and the Roman numerals are labels they’re not elements so we can ignore the Roman numerals so all the possible answers are in the box with the exception of the Roman numerals so we’re gonna use the Venn diagram to determine each of the following sets we’re gonna start off with the easier questions and after every couple of examples we’re just gonna get a little bit trickier as we go so a says set a intersected with set B you’ll notice each of the sets is represented with a circle it’s labeled so intersection our memory trick was matches matches visually means where do the circles overlap so the matches is the overlap spot so where do circles a and B overlap well they overlap right here and it’s kind of hard to see but there is a 5 a number 5 and we still unfortunately need those annoying set brackets around our answer because we’re doing a set operation so we get this set with 5 is what they have in common those sets those circles have in common this set with 5 is the final answer to a so now taking a look at B B says a with bees in parentheses now and we do have an apostrophe a compliment to apply to what’s inside the parentheses so

whenever the the apostrophe is outside parentheses it’s floating outside parentheses like this it really means to apply it to the set operation that’s inside the parentheses so let’s think it all out we just said a minute ago that the intersection means the overlap the overlap what’s in the middle of those circles however the apostrophe means what’s not in that set that it’s attached to but still in the universal set so what’s not in the intersection what’s not in the overlap is really what this means when we put it all together so if we don’t want what’s in the overlap we don’t want the 5 we want everything else that’s still in the universal box so that’s gonna be the set we got 2 & 3 that’s outside of the overlap we also got 7 11 and 13 and don’t forget about we got 17 and 19 as well again ignore the Roman numerals they’re not elements so they’re not going to be part of the final answer here so we get a really long answer we get the set with 2 3 7 11 13 17 and 19 is everything outside of the intersection let’s take a look at Part C now and I’ll change pen colors so this is a union B we got the Union you you opening up in between to set letters to capital letters is how I know it’s Union you and our memory trick for a union was to combine the two sets together it still means that so just simply combine these two circles together and list all the elements in the circles so we have a 2 3 we have a 5 and we have 7 11 and 13 after we combine the circles together and that’s going to be our final answer okay now we’re gonna look at D so question D we have a union B again but now it’s inside parentheses and we have the complement the apostrophe outside the parentheses so I’m really gonna apply it to the Union so if Union was everything inside the circles combined and complement means what’s not so putting it together what’s not inside the circles in other words it means everything outside circles a and B outside the circles so again a union B we combined the circles together which was everything inside the circles so if I want to do a complement an apostrophe to that Union the apostrophe means the opposite or what’s not inside the circle so that means what’s outside the circles so it’s outside the circles is 17 and 19 so is it hard no but you can probably guess how they’re gonna try to trip you up a little bit and then MyMathLab homework is they’re gonna constantly go back and forth between intersection and union so you’re just gonna pay extra close attention to is the u-shape opening down for intersection for matches or is the u-shape opening up for Union for combining the circles together and then it’s probably hard to see cuz it’s on the bottom of the slide so I’m gonna write it up higher but I do have another question part here and this is part II and the question part I’m going to move up is this is the hardest one here it’s a complement Union B a complement Union B so what I would recommend for this type of a question for when they give you the diagram and they have all these set operations and

the complement is attached to an actual letter rather than floating outside parenthesis is I would chop this up into two steps so what I’m gonna do because it’s kind of tricky to see it all out you’re welcome to if you’re a visual person maybe you prefer to do the whole thing in your head that’s fine but I’m gonna choose to break this harder up on up into two steps so first I’m gonna do the complement because remember you do a complement after parenthesis there aren’t any parentheses here so I’m gonna do the complement a complement means what’s not in set a but still in the universal set so everything outside of circle a would be 7 11 13 17 and 19 so here’s a complement fact let me try to write this a little bit smaller so I have more room to work so if I look at a complement to me again we said that means was outside of circle a 7 11 13 and don’t forget about the 17 and 19 that’s just a compliment now we’re gonna unite with B so I’m gonna bring down the Union you B is just a normal B there’s no apostrophe there so we just simply list everything inside Circle B so inside Circle B we have 5 7 11 and 13 and now so that was step one is I did a compliment first and I brought down set B now step two I’m going to do kind of like the last slide Union u means to combine these two sets together so between the two sets we have a five check as I go we got a seven we have an eleven we have a thirteen we have a seventeen and we have a nineteen so that’s the final answer if you break it up into two steps a compliment United with B when you combine those two sets together you get a whole bunch of elements you get five seven eleven thirteen seventeen and nineteen if you wanted to do it all visually as one step it’s definitely possible it’s just a little bit trickier you might want to kind of color code it so to get the same answer visually if I highlight in red a compliment that was everything outside of circle a so all of these elements over here is a compliment and then we’re gonna combine it with set B so I’m going to do set B in aqua so that’s everything in Circle B and this aqua highlighter here and Union means to combine the two colors together if you’re color coding it so when I combine all the elements with both with either color either the blue or the red or both either the Aqua or the red I should say or both we see that we do have a five which was part of the final answer we have seven eleven thirteen and we have 17 19 so if you wanted to color code it just think Union you’re gonna combine the two sets so any element in either color is part of the final answer if this had been an intersection symbol instead then you would only list the area that has both colors on it because intersection would mean the overlap or the matches what they have in common instead but that is either way you do it that’s the final answer for Part II there are probably going to be a couple of word problems near the end of the section 2.3 homework and if you click on the question Help button view an example this formula is going to show up I promise it’s way easier than it looks so let’s translate this formula but pretty much it’s a formula to calculate the number in the Union so that little n means number of little n always means number of something and no matter what math you’re studying so number of the cardinality so if you want the number in

the union of two sets is what this formula is designed to give us it equals first you’re gonna count up the number and set a then you are going to add it to the number and the other set set B and then after you add it you’re going to subtract the number in there intersection the number in there overlap why are we subtracting because after you add the two sets what happens is you double count all the elements that are in the overlap and so by subtracting the overlap we subtract away the extra count to get the right answer now the way I think of this fancy formula here is Union in word problems means or it means the same as or so the or number equals the number and set a I just put an A for that plus the number and set B minus intersection is the end number in word problems in math so to find the or number the Union and take the number and one set plus the number and the other set – there and number – there overlap number so seeing this formula in action example seven says according to fact monster.com among the US presidents in the White House as of 2018 anyway 26 had dogs eleven had cats and nine had both dogs and cats how many US presidents had dogs or cats in the White House please keep in mind or in this course it’s gonna always be the inclusive or meaning one or the other or both so in other words how many US presidents had just dogs had just cats or had both dogs and cats that’s what we mean by inclusive or one or the other or both so as soon as you see them asking you for the or number you see three numbers given to you these are all clues that we can use our new formula or the or number equals the number in one set plus the number in the other set – their overlap the end number so we’re solving for the or number I’m gonna bring that variable down the number in one set well it says 26 had dogs and then we hit the comma as we go back through we’re gonna use each of those numbers so 26 belongs to one set the dog owner set so it’s your choice you can plug in the 26 into either A or B it honestly does not matter so I’m just gonna go in alphabetical order and we’ll say a so a represents all the dog owners so I’m plugging in 26 into a done so I’ll slash through that now 11 had cats 11 people had cats so 11 also belongs to one set so again you could plug it into a or B since we already plugged into a I’m gonna plug it into B in other words set B represents all the cat owners so B is now 11 and don’t forget to bring down the plus symbol in between those sets the only number left is 9 had both dogs and cats it’s literally part of the phrase with the word and is why I call it the and number so 9 is the overlap 9 is the end number plug it in plug it in and bring down your minus sign it’s part of the formula so here we go now we just compute out so we’re gonna do 26 plus 11 first which gives us 37 bring down the minus 9 now we’re gonna do our 37 minus our 9 and we get a final answer of 28 but when you’re looking through these word problems when it’s asking you for whatever scenario when you see the word or in that question at the end you’re gonna be able to use this formula you’ll add together the two sets and you’ll minus the number in the overlap every time and that will give you the final final answers a twenty eight US presidents had dogs had cats or had both of those animals