Question

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

Answer 1

$A :\{1,2,3,4,5,6,7\}$
Number of elements in set B
$= n (1 \notin T )+ n (2 \in T )- n [(1 \notin T ) \cap(2 \in T )] $
$=2^6+2^6-2^5=96$
Number of elements in set $C$
$ =\{\{2\},\{3\},\{5\},\{7\},\{1,2\},\{1,4\},\{1,6\}, $
$ \{2,3\},\{2,5\},\{3,4\},\{4,7\},\{5,6\},\{6,7\} $
$ \{1,2,4\},\{1,3,7\},\{1,4,6\},\{1,5,7\},\{2,3, $
$ 6\},\{2,4,5\},\{2,4,7\},\{2,5,6\},\{3,4,6\}, $
$\{4,6,7\},\{1,2,4,6\},\{2,4,6,7\},\{2,4,6, $
$ 5\},\{3,5,7,4\},\{1,3,5,4\},\{1,5,7,4\},\{1, $
$ 2,3,5\},\{1,2,3,7\},\{1,3,6,7\},\{1,5,6,7\}, $
$ \{2,3,5,7\},\{1,5,7,2,4\},\{3,5,7,2,6\},\{1,$
$ 3, 7, 2, 4\}, \{1,4, 5, 6, 7\}, $
$ \{1,3,4,5,6\},\{1,2,3,6,7\},\{1,2,3,5,6\}, $
$ \{1,2,3,4,6,7\} $
Number of elementrrs in $C =42$
$ \Rightarrow n(B \cup C)=n(B)+n(C)-n(B \cap C) $
$ =96+42-31=107$