IIT JAM 2018 Real analysis Question (2)

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General Mathematics

Question

Solution: we discuss options one by one:

(A) if $P$, and $Q$ are compact subsets of $\mathbb{R}$ then $P$ and $Q$ both are bounded and closed subsets of $\mathbb{R}$ (By "Heine-Borel theorem"). Also we know that, 
  • Union of two bounded subsets of $\mathbb{R}$ is bounded subset of $\mathbb{R}$
  • Finite union of closed sets in $\mathbb{R}$ is again closed set in $\mathbb{R}$.
Hence, $P\cup Q$ is again bounded and closed subset of $\mathbb{R}$ and hence by "Heine Borel theorem" it is compact. So that option (A) is true.

(B) take $P=\mathbb{Q}$ and $\mathbb{Q^c}$ then both $P$ and $Q$ are nonempty disjoint subsets of $\mathbb{R}$ that are not connected, but their union is $\mathbb{R}=(-∞,+∞)$ is connected subset of $\mathbb{R}$. Hence (B) is false. 

(C) take $P=\{1,2\}$, $Q=(1,2)$ then $P$ is closed, and $P\cup Q=[1,2]$ is closed. Also both $P,Q$ are nonempty disjoint subsets of $\mathbb{R}$. But, $Q$ is not closed subset of $\mathbb{R}$. Hence (C) is false!

(D) Similarly, take $P=(-1,0)$ and $Q=[0,-1)$ then we can see (D) is fasle too. 

Hence answer is, options (B), (D), (D). 

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