How do the set of all transitive relations on a set form a lattice? I am new to discrete mathematics and only thing that i know about lattices is that to form a lattice, each pair of the set must have a lowest upper bound and greatest lower bound.

Let $X$ be a set, and let $\mathscr{L}$ be the set of transitive relations on $X$. The lattice order on $\mathscr{L}$ is $\subseteq$. If $R,S\in\mathscr{L}$, it's not hard to show that $R\cap S$ is transitive, so it's in $\mathscr{L}$. (I'll leave it to you to try to prove that; it really is a very straightforward exercise in checking the definition.) It's clearly the largest relation on $X$ that is a subset of both $R$ and $S$, so it's their greatest lower bound in $\mathscr{L}$.
Their least upper bound is a little more complicated. Let $\mathscr{T}$ be the set of all transitive relations $T$ on $X$ such that $R\cup S\subseteq T$; then it's not hard to show that $\bigcap\mathscr{T}$ is a transitive relation on $X$, i.e., that $\bigcap\mathscr{T}\in\mathscr{L}$. (In other words, it's not just the intersection of two transitive relations that is transitive: the intersection of any collection of transitive relations on a set is transitive.) Let $T_0=\bigcap\mathscr{T}$; then $T_0\subseteq T$ for each $T\in\mathscr{T}$, so $T_0$ is the smallest member of $\mathscr{L}$ containing $R\cup S$ and hence the smallest member of $\mathscr{L}$ containing both $R$ and $S$. ($T_0$ is called the transitive closure of $R\cup S$.)
