Write the specified set using the roster method givenU={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace.(M∪N′)∩(M′∪N)(M\cup N^{\prime})\cap(M^{\prime}\cup N)

(M∪N′)∩(M′∪N)={b,c,d,e,g}(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)=\left\lbrace b,c,d,e,g\right\rbrace

Write the specified set using the roster method given U = { a , b , c , d , e , f , g , h } , M = { a , b , d } , N = { b , d , f , h } U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace . ( M ∪ N ′ ) ∩ ( M ′ ∪ N ) (M\cup N^{\prime})\cap(M^{\prime}\cup N)

Welcome back. For this problem, we'd like to find M union and complement, intersect M complement union N, given our universal set U and sets M and N. So let's get started. We have two sets of parentheses here, so we're just going to start with our set of parentheses on the left, and then we'll go to our second set next. Now, our first set is M union and complement. So we want to find the set of all elements in M, in and complement, or in both. Now, let's start by actually finding N complement, because we don't actually have it yet. Now, N complement is going to be the set of all things in the universal set, but not in end. So we'll quickly need to look through the universal set and pick out the things not in N to put in N complement. We can see that A is in the universal set, but not in N, so it's in the complement. B is inside of N, but C is not, so it's in the complement, D is inside of N. So it's not in our complement, but E is not in N, so it is in our complements. F is in N, so not in our complement, but G is not, so it is in our complements, and H is inside of N. So here we have all of the elements of N complements. We have A, C, E, and G. Now that we know what N complement is, we can set about finding M union and complement, this piece here, which again is just the set including all elements in M, in N complement, and in both. Now, let's start by adding in the elements from M. We have A, B, and D. Next, we want to add in any element from any complement that we haven't already added in. So, we did already add in A, but we haven't yet added in C, E, and G. So we'll put those in our union. Now that we took care of our first set of parentheses here, we can move on to our next set of parentheses. And we notice here that we're actually going to need M complement to find M complement union N. Now, we don't have M complement, so we should find it before finding our union. M complement is just going to be the set of all things in our universal set, not in M. So, A is an M, meaning it's not in the complement, and B is also an M, but C is not, so it's in the complement. D is an M, but the remaining elements are not. So we'll have E, F, G, and H inside of M complements. Now that we have M complements, we can set about finding M complement union N, which is going to be the set of things in M complement, in N, or in both. Let's copy down the set of elements in M complement first. We have C, E, F, G, and H. Now we need to add in any element from N that we haven't added in already. We can see that B is an N, and we haven't added it yet, and similarly, D is in N, and we haven't added it yet, but F and H are already in our union, so we don't need to add them a second time. We can now close out our set. So here we have M union and complement, and we also have M complement union N, and we want to find the intersection of these two sets. So we want to find the set of elements in both of these two sets. To do this, we'll just go through the elements of M union and complement, and check to see which ones are also in M complement union N. Now, we have A in M union and complement, but it is not also in M complement union N, so it's not in the intersection. But we can see that B is in both of our sets, so it is in the intersection. Similarly, D is in both of our sets, so it is also in the intersection. We also see C in both sets, so it's in the intersection. And we see E in both sets, so it's also in the intersection. And finally, we see G in both sets, so it's also in the intersection. Since we've gone through all of the elements of M union and complement, we don't need to check anything else. So these are all the elements of M union and complement, intersect M complement union N, and they are B, D, C, E, and G. Great work.