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I'm new to mathematics, and I've been banging my head against the wall for a while over this assignment question. I'm trying to determine whether a collection $\mathcal{F}$ is necessarily an algebra, knowing that:

1) It is closed under symmetric difference for $\mathcal{A, B} \in \mathcal{F}$ and $\mathcal{A}\Delta\mathcal{B} \in \mathcal{F}$.

2) The universe is in the collection, $\mathcal{S} \in \mathcal{F}$.

I understand that by definition, an algebra is closed under finite union (countable union for $\sigma$-algebras), and that it is also closed under complementation. I assume that if this is true, there is a way to prove that using the known facts 1 and 2. I've had no luck showing this (if applicable).

Thank you in advance for your help.

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Here's a counterexample . . .

Let $\mathcal{F}$ be the set of all subsets of $\{1,2,3,4\}$ with an even number of elements.

It's easily verified that $\mathcal{F}$ is closed under symmetric difference.

However $\mathcal{F}$ is not closed under finite intersections, since, for example, $\{1,2\} \in\mathcal{F}$ and $\{1,3\} \in\mathcal{F}$ but $\{1\} \notin \mathcal{F}$.

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    Thank you, mucho helpful. I think I was too focused on proving that it was necessary that I didn't try hard enough to find a counterexample. – 2017-01-27
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    You're welcome. As a guiding principle -- stay objective. If you can't prove something, allow for the possibility that it's not true, and see if you can break it. If it doesn't break, try again to prove it. Back and forth. There might be an initial bias towards either proof or disproof, but when the initial choice runs into obstacles, that should change the initial bias, at least a little. – 2017-01-27
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    Note that for $|S| < 4$, there are no counterexamples. Unless one has the insight on what form a counterexample should take (I didn't have that insight), continuing by hand could get tedious. Lacking that insight, I had no desire to try to test the $|S| = 4$ case by hand. Instead, I wrote a computer program, and voila -- the output gave me the example that perhaps I should have found by hand. – 2017-01-27