Tessellate S.T.E.M.S. (2019) - Computer Science - College - Set 3 - Subjective Problem 2

Let $A$ be a set and $\to$ a binary relation on $A$.

We write $x \to^* y$ , if there are finitely many elements $v_1, v_2, \cdots v_n$ such that $x \to v_1 \to \cdots v_n \to y$ .

We call $\to$ good if whenever $x \to y$ and $x \to z$ for some $z \neq y$ , there is a $w$ such that $y \to w$ and $z \to w$ .

Prove that if $\to$ is good, so is $\to^*$ . Is the converse true?

This problem is a part of Tessellate S.T.E.M.S (2019)

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$