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I don't know how to prove this. The definition of the a transitive set is $$ X \text{ is transitive} \Leftrightarrow \forall y \text{ }\forall z ( (z \in y \vee z \in x) \rightarrow z\in x)) $$

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    The question in the body is just the definition of transitive – 2017-02-24
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    Have you tried to verify the definition of transitive from $\bigcup x\subseteq x$; and vice versa? – 2017-02-24
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    @AsafKaragila My idea was to take a $y \in \bigcup x$ and prove that $y$ is also in $x$. But i think my arguments are very poor. (Sorry about my english) – 2017-02-24
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    @AndresMejia Sorry, i just correct the question and put the definition that i have in my notes. – 2017-02-24
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    That is *not* how you prove a set is transitive. Write down the following sentence (complete the blanks). "Suppose that $\bigcup x\subseteq x$. We want to show that $x$ is transitive, meaning (blank). Let $y\in x$ and let $z\in y$, then by the assumption (blank), and therefore $z\in x$." – 2017-02-24
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    Why did you delete your cool other question about $\phi(t) = e^{At}$? I had a cool answer I was just about to enter! Consider undeleting it? – 2018-09-03
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    Actually it's OK; I already answered it here: https://math.stackexchange.com/questions/1489371/if-phi0-i-identity-and-that-phits-phit-phis-for-all-t-s-in-ma?rq=1 – 2018-09-03
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    Sorry @RobertLewis but a few seconds after publish it I find out that some else already ask it and I dont want to repeat questions... However the answer that i found its yours, that's a coincidence. Thanks anyway! – 2018-09-03
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    It's OK, I've had my shot at it. I'd forgotten about that old one! Cheers! – 2018-09-03
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    @RobertLewis I undelete the answer 'cause i have some questions about your proof – 2018-09-03

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