Answers:
A' = {1, 3, 4, 7, 9, 10}
B' = {1, 2, 5, 9, 10}
A' ∩ B' = {1, 9, 10}
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Step-by-step explanation
The universal set U and union symbol U look (almost) identical. It's potentially confusing.
I'll use E to represent "everything", aka universal set.
E = {1,2,3,4,5,6,7,8,9,10} = set of everything
A = {2,5,6,8}
A' = set of stuff not in A = opposite of set A = complement of set A
To form set A' we will start with set E, and then delete anything we find in set A.
We will delete 2,5,6, and 8 from set E.
We go from {1,2,3,4,5,6,7,8,9,10}
to
{1,__,3,4,__, __,7,__ ,9,10}
to
{1, 3, 4, 7, 9, 10}
Therefore,
A' = {1, 3, 4, 7, 9, 10}
These values are not found in set A.
Note how if we union A and A', we will get the universal set.
Furthermore, the intersection of A and A' is the empty set. The two opposites have nothing in common.
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Return back to set E.
E = {1,2,3,4,5,6,7,8,9,10}
Delete anything you find in set B.
We'll delete the values 3,4,6,7, and 8.
B' = {1,2,__, __,5,__, __, __ ,9,10}
B' = {1, 2, 5, 9, 10}
This is the set of values not in set B.
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We'll intersect the previous results.
Set intersection has us look to see what numbers are in both sets at the same time.
That would be the following values: 1, 9 and 10
Therefore,
A' ∩ B' = {1, 9, 10}
These values are outside both sets A and B at the same time.
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Let's list the contents of A and B
A={2,5,6,8}
B={3,4,6,7,8}
Union them together.
A U B = {2, 3, 4, 5, 6, 7, 8}
This is the set of items that are in A, in B, or both.
Now list out the set of everything.
E = {1,2,3,4,5,6,7,8,9,10}
If we deleted items found in set AUB, then we should end up with A' ∩ B' = {1, 9, 10} found earlier.
We will delete 2, 3, 4, 5, 6, 7, 8 from set E.
E = {1,2,3,4,5,6,7,8,9,10}
(A U B)' = {1,__,__,__,__,__,__,__,9,10}
(A U B)' = {1, 9, 10}
We have shown that (A U B)' = A' ∩ B' works for this example. It's not a formal proof of De Morgan's Law, but I think this example is still useful to help show what is going on.
Another way to see what's going on is to make a Venn Diagram as shown below. I'll omit the explanation how the diagram is constructed. But let me know if you have questions about it.
Notice the values 1,9, and 10 are outside both circles A and B.
Region AUB is inside either circle or both circles. Region (AUB)' is the opposite of region AUB.