Sets - Universal Set-advanced

Given the universal set $U = \{x : x \in \mathbb{N}, x \leq 10\}$ and sets $A = \{2, 4, 6, 8\}$ and $B = \{1, 3, 5, 7, 9\}$. Find $(A \cup B)'$ and $A' \cap B'$.

1. First, list the elements of $U$: $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
2. Find $A \cup B$: $A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.
3. Calculate $(A \cup B)'$: $(A \cup B)' = U - (A \cup B) = \{10\}$.
4. Find $A'$: $A' = U - A = \{1, 3, 5, 7, 9, 10\}$.
5. Find $B'$: $B' = U - B = \{2, 4, 6, 8, 10\}$.
6. Calculate $A' \cap B'$: $A' \cap B' = \{1, 3, 5, 7, 9, 10\} \cap \{2, 4, 6, 8, 10\} = \{10\}$. Therefore, $(A \cup B)' = A' \cap B' = \{10\}$.

This example verifies one of De Morgan's Laws. The complement of the union of two sets is equal to the intersection of their individual complements within the defined Universal Set.

If $n(U) = 50$, $n(A) = 20$, and $n(B) = 25$, and $n(A \cap B) = 10$, find the number of elements that are in neither $A$ nor $B$.

1. We need to find $n((A \cup B)')$.
2. First, find $n(A \cup B)$ using the formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
3. Substitute the values: $n(A \cup B) = 20 + 25 - 10 = 35$
4. Now, use the relation with the Universal Set: $n((A \cup B)') = n(U) - n(A \cup B)$
5. Substitute the values: $n((A \cup B)') = 50 - 35 = 15$ There are $15$ elements in neither $A$ nor $B$.

The number of elements in neither set is represented by the complement of the union of the two sets within the Universal Set.