We will solve each problem step by step using set operations. The universal set is:

$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

The given sets are:

$A = \{1, 2, 4, 6, 8, 10\}, \quad B = \{1, 3, 5, 7, 8, 9\}$

Step-by-step Solutions:

11. $A'$ (Complement of $A$)

The complement of $A$ consists of all elements in $U$ that are not in $A$.

$A' = U - A = \{3, 5, 7, 9\}$

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12. $A' \cup B'$

First, find $B'$, the complement of $B$:

$B' = U - B = \{2, 4, 6, 10\}$

Now, compute the union of $A'$ and $B'$:

$A' \cup B' = \{3, 5, 7, 9\} \cup \{2, 4, 6, 10\} = \{2, 3, 4, 5, 6, 7, 9, 10\}$

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13. $A' \cap B'$

Find the intersection of $A'$ and $B'$:

$A' \cap B' = \{3, 5, 7, 9\} \cap \{2, 4, 6, 10\}$

Since there are no common elements, we get:

$A' \cap B' = \emptyset$

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14. $(A \cup B)'$

First, compute $A \cup B$:

$A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} = U$

The complement of $A \cup B$ is:

$(A \cup B)' = U - U = \emptyset$

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15. $B - A$ (Elements in $B$ but not in $A$)

Subtract $A$ from $B$:

$B - A = \{1, 3, 5, 7, 8, 9\} - \{1, 2, 4, 6, 8, 10\}$

$B - A = \{3, 5, 7, 9\}$

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Final Answers:

1. $A' = \{3, 5, 7, 9\}$
2. $A' \cup B' = \{2, 3, 4, 5, 6, 7, 9, 10\}$
3. $A' \cap B' = \emptyset$
4. $(A \cup B)' = \emptyset$
5. $B - A = \{3, 5, 7, 9\}$

Would you like me to clarify any step or provide more details?

Related Questions:

1. How do you find the complement of a set in a universal set?
2. What is the difference between set union and set intersection?
3. How do you compute the difference between two sets?
4. What happens if two sets are disjoint when computing their union and intersection?
5. How do De Morgan's laws relate to set complements and intersections?

Tip:

When dealing with set complements, always ensure that you correctly identify the universal set, as it defines the elements available for comparison.