Let A = {1, 2, 3, 4, 5, 6, 7}. Define B = {T | vedclass

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(A) The total number of subsets of $A$ is $2^7 = 128$. We need to find $n(B \cup C) = n(A) - n(B^c \cap C^c)$. $B^c = \{T \subseteq A : 1 \in T 	ext{

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$107$

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(A) The total number of subsets of $A$ is $2^7 = 128$. We need to find $n(B \cup C) = n(A) - n(B^c \cap C^c)$. $B^c = \{T \subseteq A : 1 \in T 	ext{ and } 2 
otin T\}$. $C^c = \{T \subseteq A : 	ext{the sum of elements of } T 	ext{ is not a prime number (including 0)}\}$. For $T \in B^c$,$T = \{1\} \cup S$,where $S \subseteq \{3, 4, 5, 6, 7\}$. The sum of elements of $T$ is $1 + 	ext{sum}(S)$. We need $1 + 	ext{sum}(S)$ to be non-prime. The possible sums of $S$ range from $0$ to $3+4+5+6+7 = 25$. By calculating the subsets $S$ such that $1 + 	ext{sum}(S)$ is not prime,we find $n(B^c \cap C^c) = 21$. Thus,$n(B \cup C) = 128 - 21 = 107$.

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. Define $B = \{T \subseteq A : \text{either } 1 \notin T \text{ or } 2 \in T\}$ and $C = \{T \subseteq A : \text{the sum of all the elements of } T \text{ is a prime number}\}$. Then the number of elements in the set $B \cup C$ is $\dots\dots$

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