Key Concepts and Formulas

• Power Set: The set of all subsets of a given set. If a set $A$ has $n$ elements, its power set $\mathcal{P}(A)$ has $2^n$ elements.
• Principle of Inclusion-Exclusion for Sets: For two sets $B$ and $C$, $n(B \cup C) = n(B) + n(C) - n(B \cap C)$.
• De Morgan's Laws for Sets: $(B \cup C)' = B' \cap C'$ and $(B \cap C)' = B' \cup C'$.
• Complementary Counting: $n(X) = n(\mathcal{U}) - n(X')$, where $\mathcal{U}$ is the universal set.
• Prime Numbers: Positive integers greater than 1 that have no positive divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11, ...). 0 and 1 are not prime numbers.

Step-by-Step Solution

The problem asks for the number of elements in the set $B \cup C$, where $B$ and $C$ are defined based on subsets of $A=\{1,2,3,4,5,6,7\}$. We will use the complementary counting approach. The total number of subsets of $A$ is $n(\mathcal{P}(A)) = 2^{|A|} = 2^7 = 128$. We will find $n((B \cup C)')$ and subtract it from $n(\mathcal{P}(A))$. By De Morgan's Law, $(B \cup C)' = B' \cap C'$. We need to find the number of subsets $T \subseteq A$ such that $T \in B'$ and $T \in C'$.

Step 1: Characterize the set $B'$

The set $B$ is defined as $B=\{T \subseteq A : 1 \notin T \text{ or } 2 \in T\}$. The complement $B'$ consists of subsets $T$ that do NOT satisfy the condition for $B$. The negation of "($1 \notin T$) or ($2 \in T$)" is "($\neg(1 \notin T)$) and ($\neg(2 \in T)$)", which simplifies to "$1 \in T$ and $2 \notin T$". So, $B' = \{T \subseteq A : 1 \in T \text{ and } 2 \notin T\}$. To count the number of such subsets, we fix that 1 must be in $T$ and 2 must not be in $T$. The remaining elements are $\{3,4,5,6,7\}$, which has 5 elements. Each of these 5 elements can either be included in $T$ or not. Therefore, $n(B') = 2^5 = 32$.

Step 2: Characterize the set $C'$

The set $C$ is defined as $C=\{T \subseteq A : \text{the sum of all the elements of } T \text{ is a prime number}\}$. The complement $C'$ consists of subsets $T$ where the sum of elements is NOT a prime number. This includes sums that are composite, 0 (for the empty set), or 1.

Step 3: Characterize and count the elements in $B' \cap C'$

We need to find subsets $T$ such that $T \in B'$ and $T \in C'$. From Step 1, $T$ must contain 1 and not contain 2. So $T$ must be of the form $\{1\} \cup X$, where $X$ is a subset of $A \setminus \{1,2\} = \{3,4,5,6,7\}$. Let $S(T)$ be the sum of elements in $T$. Then $S(T) = 1 + S(X)$, where $S(X)$ is the sum of elements in $X$. The condition for $T \in C'$ is that $S(T)$ is NOT a prime number.

It is easier to count the number of subsets in $B'$ whose sum IS a prime number, and subtract this from $n(B')$. Let $n(B' \cap C)$ be the number of subsets $T \in B'$ such that $S(T)$ is prime. Then $n(B' \cap C') = n(B') - n(B' \cap C)$.

The set $X$ is a subset of $\{3,4,5,6,7\}$. The minimum sum $S(X)$ is $S(\emptyset) = 0$. The maximum sum $S(X)$ is $S(\{3,4,5,6,7\}) = 3+4+5+6+7 = 25$. The sum $S(T) = 1 + S(X)$ ranges from $1+0=1$ to $1+25=26$. The prime numbers in the range $[1, 26]$ are $\{2, 3, 5, 7, 11, 13, 17, 19, 23\}$. For $S(T)$ to be prime, $S(X)$ must be one of $\{2-1, 3-1, 5-1, 7-1, 11-1, 13-1, 17-1, 19-1, 23-1\}$, which are $\{1, 2, 4, 6, 10, 12, 16, 18, 22\}$.

Now, we find the subsets $X \subseteq \{3,4,5,6,7\}$ for each required sum $S(X)$:

• $S(X)=1$: No subset of $\{3,4,5,6,7\}$ sums to 1.
• $S(X)=2$: No subset of $\{3,4,5,6,7\}$ sums to 2.
• $S(X)=4$: $X=\{4\}$. (1 subset)
• $S(X)=6$: $X=\{6\}$. (1 subset)
• $S(X)=10$: $X=\{3,7\}$ or $X=\{4,6\}$. (2 subsets)
• $S(X)=12$: $X=\{5,7\}$ or $X=\{3,4,5\}$. (2 subsets)
• $S(X)=16$: $X=\{3,6,7\}$ or $X=\{4,5,7\}$. (2 subsets)
• $S(X)=18$: $X=\{5,6,7\}$ or $X=\{3,4,5,6\}$. (2 subsets)
• $S(X)=22$: The sum of $\{3,4,5,6,7\}$ is 25. To get a sum of 22, we must exclude elements summing to $25-22=3$. So $X=\{4,5,6,7\}$. (1 subset)

The total number of subsets $T \in B'$ whose sum is prime is $n(B' \cap C) = 1 + 1 + 2 + 2 + 2 + 2 + 1 = 11$.

Now we can find $n(B' \cap C')$: $n(B' \cap C') = n(B') - n(B' \cap C) = 32 - 11 = 21$.

Step 4: Calculate $n(B \cup C)$

Using the complementary counting principle: $n(B \cup C) = n(\mathcal{P}(A)) - n((B \cup C)')$ $n(B \cup C) = n(\mathcal{P}(A)) - n(B' \cap C')$ $n(B \cup C) = 128 - 21 = 107$.

Common Mistakes & Tips

• Negating Conditions: Be careful when negating compound logical conditions (OR and AND). Use De Morgan's laws correctly.
• Prime Number Definition: Remember that 1 is not a prime number. The smallest prime number is 2.
• Systematic Enumeration: When finding subsets with specific sums, be organized to avoid missing cases or counting duplicates. Grouping by the number of elements in the subset can be helpful.

Summary

We used the principle of complementary counting to find the number of elements in $B \cup C$. This involved calculating the total number of subsets of $A$, and subtracting the number of subsets that are neither in $B$ nor in $C$ (i.e., $B' \cap C'$). We characterized $B'$ as subsets containing 1 and not containing 2. Then, we found the subsets in $B'$ whose sum of elements is a prime number. By subtracting this count from the total number of subsets in $B'$, we obtained $n(B' \cap C')$, which allowed us to find $n(B \cup C)$.

The final answer is $\boxed{107}$.