Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20},G={1,2,3,4,5}U=\left\lbrace1,2,3,\ldots,20\right\rbrace,E=\left\lbrace2,4,6,\ldots,20\right\rbrace,F=\left\lbrace5,10,15,20\right\rbrace,G=\left\lbrace1,2,3,4,5\right\rbrace.(D) (E∩F′)∪(G′∩F′)\left(E\cap F^{\prime}\right)\cup\left(G^{\prime}\cap F^{\prime}\right)

(E∩F′)∪(G′∩F′)={2,4,6,7,8,9,11,12,13,14,16,17,18,19}\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)=\left\lbrace2,4,6,7,8,9,11,12,13,14,16,17,18,19\right\rbrace

Write the specified set using the roster method given <math alttext="U equals the set 1 comma 2 comma 3 comma dot dot dot comma 20 comma E equals the set 2 comma 4 comma 6 comma dot dot dot comma 20 comma F equals the set 5 comma 10 comma 15 comma 20 comma G equals the set 1 comma 2 comma 3 comma 4 comma 5" data-value="U=\left\lbrace1,2,3,\ldots,20\right\rbrace,E=\left\lbrace2,4,6,\ldots,20\right\rbrace,F=\left\lbrace5,10,15,20\right\rbrace,G=\left\lbrace1,2,3,4,5\right\rbrace"> U = { 1 , 2 , 3 , … , 20 } , E = { 2 , 4 , 6 , … , 20 } , F = { 5 , 10 , 15 , 20 } , G = { 1 , 2 , 3 , 4 , 5 } U=\left\lbrace1,2,3,\ldots,20\right\rbrace,E=\left\lbrace2,4,6,\ldots,20\right\rbrace,F=\left\lbrace5,10,15,20\right\rbrace,G=\left\lbrace1,2,3,4,5\right\rbrace . (D) ( E ∩ F ′ ) ∪ ( G ′ ∩ F ′ ) \left(E\cap F^{\prime}\right)\cup\left(G^{\prime}\cap F^{\prime}\right)

Welcome back. For this problem, we've been given you our universal set, which is a set of whole numbers from 1 to 20. We also have E, the set of even numbers from 1 to 20, F are multiples of 5 from 1 to 20, and G, the 1st 5 whole numbers in the range from 1 to 20. We want to find E intersect F complements, union G complement intersect F complements, and we know we have to start with our first set of parentheses with E intersect F complement. Now, this is going to be the set of things in E and in F complement, which is really just the set of things in E, not also in F. Because E is a set of even numbers from 1 to 20 and F is the set of multiples of 5 from 1 to 20, this is going to be all of the even numbers from 1 to 20 that are not also multiples of 5. So, going through our even numbers, we have 2. 46, and 8, all even, so in E, but not multiples of 5, so also in F complement, but we have to skip 10 here. It is an even number, but it's a multiple of 5, so it's not an F complement. Next, we have 1214, 16, and 18, all even numbers in E that are not multiples of 5, so they're an F complement as well. But the last element of E, 20, is a multiple of 5, so it's not going to be an F complement or the intersection. Here we have that E intersect F complement is 2468, 1214, 16, and 18. Next, we want to go to our second set of parentheses and find G complement intersect F complement. Again, we know that F complement is a set of things that aren't multiples of 5, in the range of whole numbers from 1 to 20. Now, G complement is going to be everything in our universal set from 1 to 20, but not in G, the range from 1 to 5. So that means that G complement is going to include all the whole numbers bigger than 5, but only up to 20, of course. So G complement is going to include 6789, 10, and so on and so forth, up to 20. So, G complement intersect F complement, the things in G complement and an F complement, is going to be the set of things from 6 to 20 that are not multiples of 5. So this is going to include 678, and 9, all whole numbers from 6 to 20 that are not multiples of 5, but it won't include 10, which is a multiple of 5, and therefore not an F complement. Next we have 1112, 13. 14 and not 15, right? Because that is a multiple of 5, not enough complement. Then 1617, 18, and 19, all in G complement from 6 to 20, but not multiples of 5, so in F complement, but of course we don't have 20, which is a multiple of 5, and therefore in F, not its complement. So we get the G complement intersect F complement, includes 6789, 1112, 1314, 1617, 18, and 19. The whole numbers in the range from 6 to 20 that are not multiples of 5. Now, finally, we want to put our pieces together, and we want to find E intersect F complement, union, G complement intersect F complement, essentially any element in one or both of our intersections. So again, this is going to be E intersect F complements, union. G complement intersect F complement. Now we have two options here. We can do what we've typically been doing, which is just go set by set, add everything from the first set, and then add everything in the second set we haven't already added, or since both sets are in numerical order, we can kind of check both sets at once, if we want to get our elements in numerical order. I'm going to do that here, but we do have to be careful when we do something like this. It's a bit quicker, but of course you don't want to miss anything. Now, we have 2, and 2 is in one of our sets, E intersect F complement, so it's in our union. We also have 4 in E intersect F complement. Both of our sets include 6, so that's in the union. Then we have 7 in G complement intersect F complements, so that's in the set. 8 is in both of our sets, so it's in our union. We have 9 in our second intersection, so it's in the union. Neither set has 10, but we have 11 inside of G complement intersect F complements. We'll add that next. Both of our sets have 12, so we'll add that. 13 is in our second intersection, so we'll add that. Then we have 14 in both of our sets, so we'll add that. We don't have 15 in either set. We have 16 in both sets, so we'll add that. We have 17 in the second intersection, so we'll put that in the union. Next, we have 18 in both of our intersections, so it's in the union, and our final element of our last intersection is 19, and so that's in the union. Now, we've gone through all of the elements of both of our intersections, and added any element that appears in one or both of those intersections to our final set union here. So, We can say that E intersect F complement, union G complement intersect F complement, is a set including 246789, 1112, 1314, 1617, 18, and 19. Great work.

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20},G={1,2,3,4,5}U=\left\lbrace1,2,3,\ldots,20\right\rbrace,E=\left\lbrace2,4,6,\ldots,20\right\rbrace,F=\left\lbrace5,10,15,20\right\rbrace,G=\left\lbrace1,2,3,4,5\right\rbrace.(D) (E∩F′)∪(G′∩F′)\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)

(E∩F′)∪(G′∩F′)={2,4,6,7,8,9,11,12,13,14,16,17,18,19}\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)=\left\lbrace2,4,6,7,8,9,11,12,13,14,16,17,18,19\right\rbrace

(E∩F′)∪(G′∩F′)={5,10,15,20}\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)=\left\lbrace5,10,15,20\right\rbrace

(E∩F′)∪(G′∩F′)={2,4,6,7,8,9,11,12,13,14,15,16,17,18,19}\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)=\left\lbrace2,4,6,7,8,9,11,12,13,14,15,16,17,18,19\right\rbrace

(E∩F′)∪(G′∩F′)={15,20}\left(E\cap $$F^{\prime}$$\right)\cup\$$left(G^{\prime}$$\cap $$F^{\prime}$$\right)=\left\lbrace15,20\right\rbrace

2. Sets

Venn Diagrams and Operations with Three Sets

Liberal Arts Math

2. Sets

Venn Diagrams and Operations with Three Sets

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Multiple Choice

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}, G = {1, 2, 3, 4, 5}$ .
(D) $(E \cap F^{'}) \cup (G^{'} \cap F^{'})$

$(E \cap F^{'}) \cup (G^{'} \cap F^{'}) = {5, 10, 15, 20}$

$(E \cap F^{'}) \cup (G^{'} \cap F^{'}) = {2, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19}$

$(E \cap F^{'}) \cup (G^{'} \cap F^{'}) = {2, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19}$

$(E \cap F^{'}) \cup (G^{'} \cap F^{'}) = {15, 20}$

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