Question

Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T\subseteq A:$ either $1\notin Tor2\in T\}$ 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

• A. 107.00
• B. 107
• C. 107.0

Answer 1

Answer: (A), (B), (C)

$\because (B\cup C)=B^{\prime }\cap C^{\prime }$

$B^{\prime }$ is a set containing sub sets of A containing element $1$ and not containing $2$. And $C^{\prime }$ is a set containing subsets of $A$ whose sum of elements is not prime.
So, we need to calculate number of subsets of $\{3,4,5,6,7\}$ 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 $\{3,4,5\},\{3,6,7\},\{4,5,6\}$ or $\{5,6,7\})$
Number of such $2$ elements subset $=7$ (except selecting $\{3,7\},\{4,6\},\{5,7\}$)
Number of such $1$ elements subset $=3$ (except selecting $\left\{4\right\}$ or $\left\{6\right\}$)
Number of such $0$ elements subset $=1$
$n(B^{\prime }\cap C^{\prime })=21\Rightarrow n(B\cup C)=2^7-21=107$