Part - A
Answer all the questions. Each question carries 5 marks.
1. Can you logically replace (p⇐⇒ q)with (p∧q)∨(¬p∧ ¬q)? Ex-
plain your answer. [Where p and q are the atomic statements]
2. Does {√n+1−√n}∞
1converge to 0? Explain your answer.
3. Let E={0} ∪{1
n|n∈N}.f(x)is defined as follows:
f(x) = (n
n+1if x=1
n,n=1, 2, 3, ....
1 if x=0
Using ϵ−δdefinition, show that fis a continuous function on E
Part - A Solutions (These are not the only ways to solve the problems.)
1. Can you logically replace (p⇐⇒ q)with (p∧q)∨(¬p∧ ¬q)? Explain
your answer. [Where p and q are the atomic statements]
Answer (Hints):
p q (p↔q)
T T T
T F F
F T F
F F T
p q (p∧q)∨(¬p∧¬q)
T T T
T F F
F T F
F F T
Yes. We can.
2. Does {√n+1−√n}∞
1converge to 0? Explain your answer.
Answer (Hints): Use the identity: x−y=x2−y2
x+y.
Therefore, |(√n+1−√n)|=1
(√n+1+√n)<1
2√n;
1
