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Normally, when you have a long equation, you can split it on two lines. Suppose that $a$ and $b$ are very long expression. Then, for example:

$$ x = a - b$$

can be rewritten as

$$ x = a + $$ $$\quad-(b)$$

I use $+$ and $-$ to maintain a consistency. What do I have to do when using boolean operator? For example, I have to split into two lines the followings:

$$x = a \wedge b$$ $$x = a \vee b$$

Which is the correct way to do this?

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1 Answer 1

up vote 2 down vote accepted

If you're trying to express conjunction $\land$ in terms of disjunction $\lor$, then you'll need to use the negation symbol $\lnot$ so you can use DeMorgan's: $$x = a\land b \iff x =\lnot(\lnot a \lor \lnot b)$$

Likewise, $$x = a \lor b \iff x = \lnot(\lnot a \land \lnot b)$$


If you're simply wanting to split lengthy propositions $a, b$ onto separate lines (for example when trying to format a paper which allows only at most a set number of characters per line, so that you cannot fit all of, say, $a \lor b$ on one line), then you can do either of the following: $$\begin{align} x &= (a) \lor \\ &\quad (b)\end{align}$$ or else $$\begin{align} x & = (a) \\&\quad \lor (b)\end{align}$$

Likewise for $x = a \land b$.

I put $a, b$ in parentheses to emphasize that the main connective is $\lor$, (or $\land$, repsectively): that we are connecting all of $a$ with all of $b$.

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the second part of your answer is what I was looking for! Thank you very much! –  the_candyman Feb 22 at 15:51
1  
You're welcome. I added the second part because after rereading your question, it occurred to me that that might be what you were after! –  amWhy Feb 22 at 15:52

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