Intereting Posts

$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
Semisimplicity is equivalent to each simple left module is projective?
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Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?
Prove that $b^r =\sup B(r)$ when $r$ is rational.
How can I prove that $x-{x^2\over2}<\ln(1+x)$
On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$
Algorithmic Analysis Simplified under Big O
How to apply Stokes' Theorem for manifolds with boundary
Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.
When are all subsets intersections of countably many open sets?
What is a counterexample to the converse of this corollary related to the Dominated Convergence Theorem?
Presentation of $D_{2n}$

$\def\p{\mathrel p}$If $\p$ is a relation on a set $A$, define $\p^2$ by $a \mathrel{\p^2} b$ if and only if there exists $c$ with $a \p c$ and $c \p b$.

If $p$ is reflexive/symmetric/transitive does $p^2$ have the same properties?

I’m not even sure how to start this, I assume I would need to use the $a$ related to $c$, $c$ related to $b$ somehow?

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Let’s do reflexivity: Suppose, $p$ is reflexive. Let $a \in A$. We want to show that $a \mathrel{p^2} a$. By the definition of $p^2$, we have to find a $c \in A$ with $a \mathrel p c$, $c \mathrel p a$. But by the reflexivity of $p$, we know that $a \mathrel p a$. So if we let $c = a$, we have $a \mathrel p c$, $c \mathrel p a$, so $a \mathrel{p^2} a$ holds and $p$ is reflexive.

I hope, this will help you to do symmetry and transitivity on your own.

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