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Why do both "and" and "or" exist?

"and" is just a change in the argument of "or", and vice versa.

$$ a \cap b = \neg(\neg a \space \cup \space \neg b) $$

$$ a \cup b = \neg(\neg a \space \cap \space \neg b) $$

So why do we have both of them? Do they both exist simply for convenience in defining other more complicated logical structures?

I suppose another way of asking it is, why do we not have no "or's" or no "and's"? (I think)

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    Maybe we should throw out and, or, and not and just use nand. – 2017-01-25
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    I vote for nor. – 2017-01-25
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    "If *and* did not exist, it would be necessary to invent it." – not Voltaire – 2017-01-25
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    If we use the S, K, and U combinator ($UPQ$ acts like $\lnot \exists x ~ P(x) \land Q(x)$) then we can get rid of all propositional symbols (including nand), all prediate logic symbols ($\forall$, $\exists$), and all function notation. – 2017-01-25

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First of all, in intuitionistic logic those equivalences don't hold.

But even inside classical logic, there's good reason to have symbols for each: they're quite common! (We also have symbols "$\implies$" and "$\iff$" for the same reason.) Having symbols for everything isn't great, but using the bare minimum of symbols isn't great either; you strike a balance. "And," "or," "implies," "iff," and "not" are really commonly used connectives in mathematics, so we've found it useful to have symbols for each of them.

That said, in proofs, we often do reduce to a "minimal-symbols" case: e.g. if you're proving something by induction on the complexity of some formula. So just as having more symbols is sometimes useful, using the bare minimum of symbols is also sometimes useful.