Let A = \ 1, 2, 3, 4, 5, 6, 7\ . Define B = T ⊆ A : either 1 T or 2...

We know that,

By demorgan's law ${\left(B \cup C\right)}^{'} = {B}^{'} \cap {C}^{'}$

${B}^{'}$ is a set containing sub sets of A$containing element$1$and not containing$2$.</p><p>And${C}^{'}$is a set containing subsets of A$ whose sum of elements is not prime.

So, we need to calculate number of subsets of $\left\{3, 4, 5, 6, 7\right\}$ whose sum of elements plus $1$ is composite.

Number of such $5$ elements subset $= 1$

Number of such $4$ elements subset $= 3$ (except selecting $3 or 7$)

Number of such $3$ elements subset $= 6$ (except selecting \left\{3, 4, 5\right\}, \left\{3, 6, 7\right\}, \left\{4, 5, 7\right\}$or \left\{5, 6, 7\right\})</p><p>Number of such$2$elements subset$= 7$(except selecting \left\{3, 7\right\}, \left\{4, 6\right\}, \left\{5, 7\right\})</p><p>Number$ of such1$elements subset$= 3$(except selecting \left\{4\right\}$ or \left\{6\right\})

Number of such$0$ elements subset $= 1$

So, number of elements in $n\left({B}^{'} \cap {C}^{'}\right)$ = 1 + 3 + 6 + 7 + 3 + 1 = 21$</p><p>And number of elements in$n$\left(B \cap C\right)$ = {2}^{7} - 21 = 107$</p></blockquote>$