Question text is in black, answers in blue.
For each of the six parts, answer "not a proposition", "a true proposition", or "a false proposition". Part (c) requires some basic knowledge about insects which can be found on Wikipedia -- I recommend this article, for example, near the bottom of the page.
Yes, all the rules are tautologies, and always being true is exactly what makes a compound statement a tautology.
What about "p ∧ ¬ p"? That's always false, right?
Yes, if a statement is always false then its negation is a tautology. You could prove "¬(p ∧ ¬ p)" easily, either by a truth table or by applying DeMorgan to the Excluded Middle tautology above. So this means that in a proof, if you have a statement of the form "p ∧ ¬p" you may replace it with "0", and if you have a statement of the form "p ∨ ¬p" you may replace it with "1", even if "p" is some compound statement.
Now I've got a statement of the form "u ∧ v ∧ w ∧ x ∧ y", and I want to separate out "y ∧ u". Can I do that?
Yes. The AND operation is commutative and associative, so when you have a lot of things all ANDed together you may reorder them and parenthesize them however you like (and then, in this case, use Left or Right Separation) to isolate the pieces that you want). But remember that you need all ANDs, all ORs, or all XORs to do this, not a mixed combination and not →'s.
Last modified 22 September 2010