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Can anyone help me, please, solve these problems below?

1. Find the relations $R, S$ on the same set $X$, so that
$R∘S \neq S∘R.$

2. Find the relation $R$ on the suitable set $X$, so that
$\forall n \in \mathbb N: R^n \neq R^{n+1}$. The expression $R^n$ means n-times composed relation $R$,
i.e. $R^n = R∘R∘...∘R$ (n-times).

Edit: I am not really sure what should be the output of these problems. If I should write a general formula of the reations or I can just name one example.

Alex M.
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Michelle
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1 Answers1

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HINT: For the first question let $X=\{0,1\}$, and try to find relations $R$ and $S$ on $X$ such that $R\circ S=\{\langle 0,0\rangle\}$ and $S\circ R=\{\langle 1,1\rangle\}$.

For the second, find a set and a relation on that set whose associated digraph looks like this:

$$\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\ldots$$

Brian M. Scott
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