Continuing the convention above, let $\backslash$pi $,\backslash$sigma be two partitions in $\backslash$Pi n$.$ Define a relation

$$ on $\{1,2,...,$ n$\}$ by declaring a $$ b whenever there exists a sequence of elements

a $=$ e$0,$ e$1,...,$ ek $=$ b $($k can be equal to $0)$ such that for each i $=1,...,$ k$,$ either

ei$1,$ ei are in the same part of $\backslash$pi or in the same part of $\backslash$sigma $($or both$).$ Show that

$(1)$ is an equivalent relation, thus inducing a partition $$ itself;

$(2)>=\backslash$pi $,\backslash$sigma ; and

$(3)$ whenever $\backslash$eta $>=\backslash$pi $,\backslash$sigma $,$ we must have $<=\backslash$eta $.$