Guide to Negating Formulas, Study notes of Logic

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Guide to Negating Formulas

Hi everybody!

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!

For example.

Let's imagine that you want to negate this formula to the left.

¬ 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 ))

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 ))

I'm going to write this rules up at the top of the screen.

¬(¬ p ∧ ( q ∨ r ))

¬( A ∨ B )

¬ A ∧ ¬ B

¬( A ∧ B )

¬ A ∨ ¬ B

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.

¬(¬ p ∧ ( q ∨ r ))

¬( A ∨ B )

¬ A ∧ ¬ B

¬( A ∧ B )

¬ A ∨ ¬ B

¬(¬ p ∧ ( q ∨ r ))

¬( A ∨ B )

¬ A ∧ ¬ B

¬( A ∧ B )

¬ A ∨ ¬ B

A rule like this means “if you see Before , replace it with After .”

Before

After

So let's look at this rule on the right.

¬(¬ p ∧ ( q ∨ r ))

¬( A ∨ B )

¬ A ∧ ¬ B

¬( A ∧ B )

¬ A ∨ ¬ B