cmi-entrance 2012 QB9

cmi-entrance · India · ugmath 10 marks Proof Proof That a Map Has a Specific Property

Let $N$ be the set of non-negative integers. Suppose $f : N \rightarrow N$ is a function such that $f ( f ( f ( n ) ) ) < f ( n + 1 )$ for every $n \in N$. Prove that $f ( n ) = n$ for all $n$ using the following steps or otherwise. a) If $f ( n ) = 0$, then $n = 0$. b) If $f ( x ) < n$, then $x < n$. (Start by considering $n = 1$.) c) $f ( n ) < f ( n + 1 )$ and $n < f ( n + 1 )$ for all $n$. d) $f ( n ) = n$ for all $n$.

Let $N$ be the set of non-negative integers. Suppose $f : N \rightarrow N$ is a function such that $f ( f ( f ( n ) ) ) < f ( n + 1 )$ for every $n \in N$. Prove that $f ( n ) = n$ for all $n$ using the following steps or otherwise.\\
a) If $f ( n ) = 0$, then $n = 0$.\\
b) If $f ( x ) < n$, then $x < n$. (Start by considering $n = 1$.)\\
c) $f ( n ) < f ( n + 1 )$ and $n < f ( n + 1 )$ for all $n$.\\
d) $f ( n ) = n$ for all $n$.

Paper Questions

QA1 QA2 QA3 QA4 QA5 QB1 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9