Hi. I have this problem.

Let $\displaystyle \theta$ be a binary relation on a set $\displaystyle A$, which is not necessarily transitive.

Define a binary relation $\displaystyle \theta^*$ on $\displaystyle A$ as follows: for any $\displaystyle a,b \in A$, where $\displaystyle k \ge 2$ is an integer (which relies on $\displaystyle a$ and $\displaystyle b$), such that $\displaystyle a=a_1, b=a_k$ and $\displaystyle a_i \theta a_{i+1}$ for all $\displaystyle i=1,2,...,k-1$. $\displaystyle \theta*$ is called the transitive closure of $\displaystyle \theta$.

Prove that $\displaystyle \theta^*$ is a transitive relation on $\displaystyle A$.

This is what I did... but not sure if it really make sense though.

Suppose $\displaystyle <a_1, a_2>$ and $\displaystyle <a_2,a_k>$ are in $\displaystyle \theta^*$.

Show that $\displaystyle <a_1,a_k>$ is in $\displaystyle \theta^*$.

By definition of $\displaystyle \theta^*$,

$\displaystyle <a_1, a_2>$ is in $\displaystyle \theta^m$ for some $\displaystyle m$

$\displaystyle <a_2,a_k>$ is in $\displaystyle \theta^n$ for some $\displaystyle n$

Then $\displaystyle <a_1, a_k>$ is in $\displaystyle \theta^m\theta^n=\theta^{m+n}$ which is contained in $\displaystyle \theta^*$. Hence $\displaystyle \theta^*$ must be transitive. i.e. $\displaystyle \theta^*$ is a transitive relation on $\displaystyle A$.

What I'm not sure what I have done is choosing an artribrary number in the sequence between $\displaystyle a$ and $\displaystyle b$ is correct, it is where I chose $\displaystyle a_2$. Also shall I use $\displaystyle a_1$ and $\displaystyle a_k$ instead of $\displaystyle a$ and $\displaystyle b$ in my proof? Or should i just stick with $\displaystyle a$ and $\displaystyle b$?

Thanks heaps for your help.