Hello aman_cc Originally Posted by

**aman_cc** ...

2. **Prove $\displaystyle S' \cong J_m$ where $\displaystyle m<n$**

...

**Problem I am facing**: Step 2 above. How do I prove it formally. I know it sounds so much true from intuition but how do I show it in the language of formal mathematics.

I'm not sure about Part 2, but can't you say something like this for Part 1, Step 2?

Suppose that $\displaystyle f : S\mapsto J_n$ and $\displaystyle f:S'\mapsto J_m$ where $\displaystyle m \le n$. We must prove $\displaystyle m < n$.

$\displaystyle S'$ is a proper subset of $\displaystyle S \Rightarrow \exists p \in S$ for which $\displaystyle p \notin S'$

Then $\displaystyle f(p)\notin f(S')$, for otherwise $\displaystyle f^{-1}\Big(f(p)\Big) \in S'$.

Further$\displaystyle \forall r \in S'$

$\displaystyle S' \subset S \Rightarrow r \in S$

$\displaystyle \Rightarrow f(S')\subseteq f(S)$

But we have now found a $\displaystyle f(p) \in f(S)$ for which $\displaystyle f(p) \notin f(S')$. Hence $\displaystyle f(S') = J_m$ is a proper subset of $\displaystyle f(S)=J_n \Rightarrow m < n$.

Grandad