# xRy if and only if x is a descendant of y, on the set of all humans. Explain the relations

xRy if and only if x is a descendant of y, on the set of all humans.

I have the solution to this. I just don't understand how transitivity follows.

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If $x$ is a descendant of $y$ and $y$ is a descendant of $z$, then $x$ is a descendent of $z$. In other words: $xRy$ and $yRz$ implies $xRz$.

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If your father (y) is a descendent of your grandfather (z) then yRz. If you (x) are a descendent of your father then xRy. You are also, by transitivity of the relation, a descendent of your grandfather, thus xRz.

Notice that the R relationship behaves similar to inequality in respect to transitivity and reflectivity.

Transitivity:

$$2 < 3 \hspace{1in} xRy$$

$$3 < 5 \hspace{1in} yRz$$

$$\Rightarrow 2 < 5 \hspace{.5in} \Rightarrow xRz$$

But $2 < 3 \not\Rightarrow 3 < 2$, likewise $xRy \not\Rightarrow yRx$.

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