De Morgan's laws and a printout of all Boolean algebra laws
Introduction
There are a number of rules in Boolean algebra that help us derive and simplify statements. Together these are known as De Morgan's rules. In this section, we will discuss De Morgan's rules. We will see how they are applied in another section, with lots of examples. One use for them that you will discover in another section is that they can be used to simplify NAND and NOR gates in logic gate designs.
An easy way to remember De Morgan's rules is that each term is complemented, and
then the ORs become ANDs, and the ANDs become ORs.
Before we start looking at De Morgan's rules, we need to know two terms, Conjunction and Disjunction.
Conjunction
Conjunction is very similar to the logic operator AND. If all inputs are true then the output is true. Else, the output is false. The Boolean algebra symbol for conjunction is Λ.
Disjunction
Disjunction is very similar to the logic operator OR. If any of the inputs are true, then the output is true. Else, the output is false. The Boolean algebra symbol for conjunction is V.
When we talk about something being 'true', we can also talk about if being a 'one' or being 'high'. When we talk about something being 'false', we can also talk about a 'zero' or a 'low'. It is important to recognise that these terms are used interchangeably both in our notes as well as in books and online so it is something to look out for.
De Morgan's rules
1) The negation of a conjunction is the disjunction of the negation.
This is quite a sentence! Perhaps it might be better to say:
the complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variables.
The following three lines all say the same thing:
-
- the compliment of (A AND B) is the equivalent to the compliment of A OR the complement of B.
- (A AND B)' ≡ (A)' OR (B)'
- (A Λ B)' ≡ (A)' V (B)'
When working with Boolean algebraic equations, by far the most popular way of expressing this equality would be:
-
- AB ≡ A + B
According to OCR's specification, the symbol they will use for the compliment is Λ so we will try to use this as much as possible, too. At the moment, this rule isn't going to mean much, but if we have a look at our first example, we can see this rule being applied and that it does indeed work.
We have two inputs in our expression, which we will call A and B. We'll put the different combinations in the first two columns. The 'negation of the conjunction' is (A Λ B)' and this will go in the third column. The 'disjunction of the negation' is in the fourth column. Working through the expressions in the third and fourth column, you can see that they are indeed producing the same outputs so De Morgan's first rule works.
2) The negation of the disjunction is the conjunction of the negation.
This is quite a sentence! Perhaps it might be better to say:
the complement of two or more ORed variables is equivalent to the AND of the complements of the individual variables
The following three lines all say the same thing:
-
- The complement of (A OR B) is the equivalent to the complement of A AND the complement of B.
- (A OR B)' ≡ (A)' AND (B)'
- (A V B)' ≡ (A)' Λ (B)'
When working with Boolean algebraic equations, by far the most popular way of expressing this equality would be:
-
- A + B ≡ A . B
According to OCR's specification, the symbol they will use for the disjunction is V so we will try to use this as much as possible, too. At the moment, this rule isn't going to mean much, but if we have a look at our first example, we can see this rule being applied and that it does indeed work.
We have two inputs in our expression, which we will call A and B. We'll put the different combinations in the first two columns. The 'negation of the disjunction' is (A V B)' and this will go in the third column. The 'conjunction of the negation' is in the fourth column. Working through the expressions in the third and fourth column, you can see that they are indeed producing the same outputs so De Morgan's second rule works, too. Remember what we said at the start:
An easy way to remember De Morgan's rules is that each term is complemented, and
then the ORs become ANDs, and the ANDs become ORs.
Summary of a all Boolean algebra laws
You can download and print out a summary sheet of all the laws and identities we have covered. It includes the 12 basic laws, the commutative, associative and distributive laws and De Morgan's laws. You should always have a copy of this summary on your table to refer to when working on Boolean algebra, until you know them all by heart!