Guide to Negating Formulas Hi everybody! We spent a little bit of time in class talking about how to negate formulas in propositional or frst-order logic. This is a really valuable skill! If you ever need to write a proof by contradiction or a proof by contrapositive, you'll need to know how to negate formulas. While this might seem a bit tricky at frst, the good news is that there's a nice, mechanical way that you can negate formulas! There's still a bit of art to it, but by learning a few simple rules and how to apply them, you can negate just about anything! For example. ¬p ∧ (q ∨ r) Let's imagine that you want to negate this formula to the left. (¬p ∧ (q ∨ r)) To do so, we're going to begin by surrounding the formula in parentheses. ¬(¬p ∧ (q ∨ r)) And putting a negation symbol in front. ¬(¬p ∧ (q ∨ r)) Technically speaking, this formula is the negation of the original formula, though it's hard to see exactly what this formula says. ¬(¬p ∧ (q ∨ r)) Most of the time, when you need to fnd the negation of a formula, you're going to want to simplify it by pushing the negations inward. ¬(¬p ∧ (q ∨ r)) The good news is that there are a number of rules we can use to do this. ¬(¬p ∧ (q ∨ r)) If you remember from our lecture on propositional logic, we saw a series of rules for simplifying negations. ¬(¬p ∧ (q ∨ r)) For example, we saw de Morgan's laws, which say that ¬(A ∧ B) ≡ ¬A ∨ ¬B ¬(A ∨ B) ≡ ¬A ∧ ¬B ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬(¬p ∧ (q ∨ r)) I'm going to write this rules up at the top of the screen. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬(¬p ∧ (q ∨ r)) We haven't talked about this notation before, but the good news is that it's not too bad. Let's take a look at this. ¬(A ∨ B) ¬(A ∧ B) Generally speaking, these rules ¬A ∧ ¬B ¬A ∨ ¬B look like this: Before After ¬(¬p ∧ (q ∨ r)) ¬(A ∨ B) ¬(A ∧ B) A rule like this means “if you see ¬A ∧ ¬B ¬A ∨ ¬B Before, replace it with After.” Before After ¬(¬p ∧ (q ∨ r)) ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B So let's look at this rule on the right. ¬(¬p ∧ (q ∨ r)) ¬(A ∨ B) ¬(A ∧ B) This rule says ¬A ∧ ¬B ¬A ∨ ¬B “if you see something of the form ¬(A ∧ B) replace it with ¬A ∨ ¬B.” ¬(¬p ∧ (q ∨ r)) ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬(¬p ∧ (q ∨ r)) Take a look at our original formula. If you look at it closely, you'll see that it matches this particular pattern. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬(¬p ∧ (q ∨ r)) That means that we can apply this rule and rewrite it. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) .so that it looks like this. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) Notice that we've “pushed” the negation deeper into the formula. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) Basically, we just need to keep applying these rule templates over and over until the formula is as simple as it can get. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) So let's focus on this part of the formula for now. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) Notice that it matches this particular template. ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ ¬(q ∨ r) So let's apply this rule to simplify the formula! ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ (¬q ∧ ¬r) Now we've got this, which has the negation pushed deeper into the formula! We're making progress! ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ (¬q ∧ ¬r) So what about the other part of this formula? ¬(A ∨ B) ¬(A ∧ B) ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ (¬q ∧ ¬r) Well, this is a double-negation. Intuitively, we'd expect these two negations to cancel out. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ (¬q ∧ ¬r) Indeed, that's what happens. We can write this down as the rule ¬¬A A ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B ¬¬p ∨ (¬q ∧ ¬r) So let's go and apply this rule! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Tada! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) At this point we can't push the negations any deeper into the formula. They're directly applied to propositions, which can't be simplifed. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) So we're done! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Of course, before moving on, we should be sure to check our work! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Here's the original formula. (Think of this as the “before” in ¬p ∧ (q ∨ r) the “before and after” shots.) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Is there some way we can check whether we negated this formula ¬p ∧ (q ∨ r) properly? ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Yes, actually! There are several. ¬p ∧ (q ∨ r) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Let's begin with an intuitive check. What do each of these formulas ¬p ∧ (q ∨ r) actually mean? ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) This formula means “p is true, ¬p ∧ (q ∨ r) or both q and r are false.” ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) This formula means “p is false and at least one of q and r ¬p ∧ (q ∨ r) is true.” ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Intuitively, these statements seem ¬p ∧ (q ∨ r) to check out as opposites. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) If the frst one is true, then either p is true (so ¬p is false), ¬p ∧ (q ∨ r) or q and r are false (so q ∨ r is false.) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Similarly, if the second one is true, then ¬p is true (so p is ¬p ∧ (q ∨ r) false), and at least one of q and r are true (so ¬q ∧ ¬r is false). ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) So in that sense, intuitively, ¬p ∧ (q ∨ r) everything checks out. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Of course, a lot of things that make intuitive sense aren't right, ¬p ∧ (q ∨ r) so perhaps we should check this in a diferent way. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) When we're dealing with propositional logic, we can always check the truth ¬p ∧ (q ∨ r) tables for these formulas and see how they compare! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) If you punch in these two formulas, here's the truth tables ¬p ∧ (q ∨ r) that you'll get back. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Notice that, going row by row, the truth values are opposites ¬p ∧ (q ∨ r) of one another. That means that they're negations of one another! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) There's actually a diferent way to use the truth table tool to ¬p ∧ (q ∨ r) check if two formulas are negations of one another. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Imagine that you have two formulas A and B. If A is a ¬p ∧ (q ∨ r) negation of B, then any time A is true, B is false and vice-versa. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) That means that the formula A ↔ B ¬p ∧ (q ∨ r) should always evaluate to false. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) So in our case, we can try punching in this formula into ¬p ∧ (q ∨ r) the truth table tool. ((pp ∨ ∨ (¬ (¬qq ∧ ∧ ¬ ¬rr)))) ↔ ↔ (¬ (¬pp ∧ ∧ ( (qq ∨ ∨ r r)))) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) If we do, here's what we ¬p ∧ (q ∨ r) get back. ((pp ∨ ∨ (¬ (¬qq ∧ ∧ ¬ ¬rr)))) ↔ ↔ (¬ (¬pp ∧ ∧ ( (qq ∨ ∨ r r)))) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B p ∨ (¬q ∧ ¬r) Since the formula is always false, we know that the two ¬p ∧ (q ∨ r) formulas must be negations of one another. Nifty! ((pp ∨ ∨ (¬ (¬qq ∧ ∧ ¬ ¬rr)))) ↔ ↔ (¬ (¬pp ∧ ∧ ( (qq ∨ ∨ r r)))) ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B That wasn't so bad! Let's do another one. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B (p ∨ q) ∧ (¬p ∧ ¬q) Let's try taking the negation of this formula. ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B (p ∨ q) ∧ (¬p ∧ ¬q) Actually, before we do this together, why don't you try doing it yourself frst! ¬¬A ¬(A ∨ B) ¬(A ∧ B) A ¬A ∧ ¬B ¬A ∨ ¬B (p ∨ q) ∧ (¬p ∧ ¬q) Take a minute or two to walk through this one on your own.