TIFR2010-Maths-B-6

Question

Let $A, B$ be subsets of $\mathbb{R}$. Define $A + B$ to be the set of all sums $x +y$ with $x \in A$ and $y \in B$. Which of the following statements is false?

• A. If $A$ and $B$ are bounded, then $A + B$ is bounded
• B. If $A$ and $B$ are open, then $A + B$ is open
• C. If $A$ and $B$ are closed, then $A + B$ is closed
• D. If $A$ and $B$ are connected, then $A + B$ is connected

Answer 1

Option $C$ is false . If $A$ and $B$ are closed then $A+B$ is closed .

Ex:$A=\{n:n= .2,3,...\},B=\{-n+\frac{1}{n}:n=2,3,..\}$ now $A+B=\{\frac{1}{n}:n=2,3,...\}$Now the limit point of this set is $0$ which  is not in this set

Answer 2

"If A and B are connected, then A + B is connected."  connected meaning in SET theory :-P ??
it must be false.

Answer 3

Answer will be B.

Connected means the set cannot be partitioned, which is true for real numbers

http://mathworld.wolfram.com/ConnectedSet.html

Comments

why is b false here plz explain