Question 1195639: Let S={1,2,3,4,5,6,7,8,9,10} be the universal set.
Let sets A, B, and C be subsets of S , where:
Set A={1,6,7,10}
Set B={2,9,10}
Set C={3,4,5,7,8}
LIST the elements in the set A∩B∩C:A∩B∩C= {___}
LIST the elements in the set A∪B∪C:A∪B∪C= {__}
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
∩ is the intersection symbol
U is the union symbol
Set A∩B is where we look at elements in both sets A and B at the same time. This is the overlapped region in the Venn Diagram.
Similarly, A∩B∩C looks at numbers in all three sets at the same time.
Set A={1,6,7,10}
Set B={2,9,10}
Set C={3,4,5,7,8}
The value 1 is in set A, but not in any other set
6 is in set A, but not in any other set
7 is in A and C, but not set B, so this value is crossed off as well
10 is in A and B, but not in set C
The values 1,6,7 and 10 are not found in all three sets.
As you can see, we only need to check one set to determine A∩B∩C since this intersected set must consist of values from that particular single set mentioned.
In other words, there isn't any value that is in all three sets at the same time.
Therefore, A∩B∩C = { } is the empty set.
We write two curly braces with nothing between them.
Not even 0 is in this set.
The set { } is different from { 0 }
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For the next part, we'll be doing set union.
A∪B means we combine sets A and B into one bigger set.
Toss out any duplicates.
Example:
A = {1,2}
B = {3,4}
A∪B = {1,2 3,4}
A∪B = {1,2,3,4}
The spacing is intentional to show how sets A and B combine together. When writing the actual union, you'd just write a list of numbers like normal without the massive gap.
Another example:
A = {1,2,3}
B = {3,4,5}
A∪B = {1,2,3 3,4,5}
A∪B = {1,2,3 3,4,5}
A∪B = {1,2,3,4,5}
We combine the 1,2,3 with 3,4,5
Cross off the second copy of 3 to avoid duplicates.
Going back to the original problem
Set A={1,6,7,10}
Set B={2,9,10}
Set C={3,4,5,7,8}
We have
A∪B∪C = {1,6,7,10, 2,9,10, 3,4,5,7,8}
A∪B∪C = {1,2,3,4,5,6,7,8,9,10}
Common practice is to sort the values in the set from smallest to largest.
Don't forget to erase any duplicates.
It turns out that we have the universal set S = {1,2,3,4,5,6,7,8,9,10} as the result of the union of all three sets mentioned.
All of the values in the universe are found in set A, set B, or set C.
We have this interesting contrast going on.
A∩B∩C is the empty set, aka "nothing"
A∪B∪C consists of everything in the universal set.
This won't always happen. It is possible your teacher purposefully crafted these sets in this fashion to attain this dichotomy.
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Answers:
A∩B∩C = { } which is the empty set
A∪B∪C = {1,2,3,4,5,6,7,8,9,10} which in this case is the universal set
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