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Q.
Let $P$ be the set of prime numbers and $S=\left\{t: 2^t-1\right.$ is a prime $\}$. Then,
Sets
Solution:
(a) Let $x \notin P$
$\rightarrow x$ is a composite number.
Let us now assume that $x \in S$
$\Rightarrow 2^x-1 =m(\text { where } m $ is a prime number)
$\Rightarrow 2^x =m+1$
which is not true for all composite numbers, say for $x=4$ because $2^4=16$ which cannot be equal to the sum of any prime number $m$ and 1 .
Thus, we arrive at a contradiction
$\Rightarrow x \notin S$
Thus, when $x \notin P$, we arrive at $x \notin S$.
So, $ S \subset P$