TIFR-2014-Maths-A-1

Question

Let $A, B, C$ be three subsets of $\mathbb{R}$. The negation of the following statement For every $\epsilon  > 1$, there exists $a \in A$ and $b \in B$ such that for all $c \in C, |a − c| < \epsilon$ and $|b − c| > \epsilon$ is 

• A. There exists $\epsilon \leq 1$, such that for all $a \in A$ and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$
• B. There exists $\epsilon \leq 1$, such that for all $a \in A$ and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ and $|b − c| \leq \epsilon$
• C. There exists $\epsilon  > 1$, such that for all $a \in A$ and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ and $|b − c| \leq \epsilon$ 
• D. There exists $\epsilon > 1$, such that for all $a \in A$ and $b \in B$ there exists $c \in C$ such that $|a − c| \geq \epsilon$ or $|b − c| \leq \epsilon$ 

Answer 1

c will be the answer