How do I convince students to use Boolean algebra? I have taught Boolean algebra in an upper-level discrete math course for CS majors. I showed them how, using just the axioms (distributive property, etc.), you can derive other properties (such as De Morgan's Law). Then I assigned them to try it, and most of them happily (or grumpily) worked through the algebra. But one student proved the statement exhaustively using a truth table, instead of using algebra. That is not what I had wanted him to do, but I was at a loss on how to justify it to him. Since Boolean variables can only have 2 different values, proving a Boolean algebra equality can always be done exhaustively. Or am I wrong?
Well, for one, computing a truth table doesn't scale: generating one takes exponential time, with respect to the number of variables you're using.

You can perhaps "subtly" drive this idea home by asking your students to manipulate some proposition that contains 7 or 8 variables (which would require creating a truth table with 128 or 256 rows). That would require an exhausting amount of work -- the benefits of using algebra should hopefully be more obvious, then.

To make the problem more tractable for the people who are using algebra, you could perhaps engineer the original statement so some of the variables end up "vanishing" (e.g. they simplify into something like p || !p).

I think it would also be worth explaining why it's useful to be able to systematically simplify or manipulate boolean expressions in the first place -- for example, you could explain that when you're designing circuits, you have a vested interest in minimizing the number of gates used to keep costs down (as well as potentially minimizing the number of unique gates used).

And finally, another idea is to just simply tell your students why you're asking them to learn this material in the first place. For example, you could say that you're trying to prepare them to write proofs later on in the course (assuming that's what you're doing). After all, a large part of writing proofs is about correctly manipulating and applying definitions, and that's exactly what you're asking the students to do now.

Generally, I think sharing your gameplan is a good way of getting students on-board (albeit sometimes reluctantly).