Applying De Morgan’s Laws
Let’s begin this lesson by learning about the mathematician Augustus De Morgan. This link will take you to a brief biography, so check it out.
Now it’s time to review De Morgan’s Laws for Logical Statements.
Suppose p and q are any two statements. Then the following statements are equivalent:
1) ~(p ∨ q) ≡ ~p ∧ ~q (The negation of p or q is equivalent to not p and not q.)
2) ~(p ∧ q) ≡ ~p ∨ ~q (The negation of p and q is equivalent to not p or not q.)
How do you apply De Morgan’s laws? The purpose of this lesson is to learn how to apply them to negate disjunctions and conjunctions. The first law is used to negate disjunctions while the second one is used to negate conjunctions. Remember that disjunctions include the connective “or,” and that conjunctions include the connective “and.”
Let’s try a couple of examples.
First look at Example 1.
Negate the statement: She is Queen Elizabeth, or he is President Obama. Is this statement true or false when we look at the photo? Yes. It is true. She is Queen Elizabeth. He is not President Obama.
How do we negate the statement? First we notice that the statement is a disjunction because of the word “or. “ That means we want to use Law 1. That gives us the following result:
She is not Queen Elizabeth, and he is not President Obama. Is this statement true or false when we look at the photo? That’s correct. It is false.
Now let’s move on to Example 2.
Negate the statement: The steaks are on the grill, and the iced tea is on the table. Based on our two photos it is true.
How do we negate the statement? This time we notice that we have a conjunction because of the word “and.” That means we want to use Law 2. That gives us the following result:
The steaks are not on the grill, or the iced tea is not on the table. As we can see, this statement is false when we look at the photos.
That concludes our brief lesson on applying De Morgan’s laws.