I'm stuck on this question and would appreciate the solution, thanks!
Let $R$ and $S$ be the relations on $A = \{1,2,3,4,5\}$ given by$$R = \{(1,2),(2,3),(3,4),(4,5)\}\\S=\{(2,3),(2,4),(3,4)\}$$Compute $R^2 = R \circ R$ and $S^2 = S\circ S$. Check that $S^2 \subset S$ while $R^2 \not\subseteq R$
You could draw the relations on a piece of paper to make it clearer:
R
$$ 1\to 2\to 3\to 4\to 5 $$ and
S
$$ \begin{array}{cc} &&3\\ &\nearrow&\\ 2&&\downarrow\\ &\searrow&\\ &&4 \end{array} $$ and then squaring them means to form every relation that can be achieved by two consecutive arrows, so $$ R^2=\{(1,3),(2,4),(3,5)\} $$ and $$ S^2=\{(2,4)\} $$ Of course this can be found out without drawing anything. The graphical illustration only goes to strengthen your concept of what it actually means, I hope.
So in this diagram form the definition says that $x\to y\in R^2$ iff there exists $z$ such that $x\to z\to y\in R$ so for instance $$ 1\to3\in R^2\quad\text{since}\quad 1\to2\to3\in R $$ In the usual format this is written as $(1,3)\in R^2$ since we have $(1,2),(2,3)\in R$ or generally speaking $(x,y)\in R^2$ iff there is some $z$ such that $(x,z),(z,y)\in R$.