Gavin Howe
De Morgan's laws are a set of two postulates widely used in set theory and Boolean algebra. They are critical in understanding mathematical arguments and simplifying complex boolean expressions. In set theory, De Morgan's laws define the relationship between the union, intersection, and complements of sets. In Boolean algebra, they give the relation between AND, OR, and NOT. De Morgan's laws can be used to simplify logical expressions in computer programming, making code more efficient and easier to verify. They are also applied in electronic engineering for developing logic gates, where they can be used to manipulate the types of logic gates and simplify operations.
| Characteristics | Values |
|---|---|
| Definition | De Morgan's law gives the relation between union, intersection, and complements in set theory. |
| Boolean Algebra | In Boolean Algebra, De Morgan's law gives the relation between AND, OR, and complements of the variable. |
| Logic | In logic, De Morgan's law gives the relation between AND, OR, or negation of the statement. |
| Applications | De Morgan's law is used in computer programming to simplify logical expressions, verify codes, and optimize code. It is also used in digital circuit design, formal logic, and electronic engineering for developing logic gates. |
| Set Theory | De Morgan's law states that the union and intersection of sets interchange under complementation. |
| First Theorem | The first theorem states that two or more variables NOR'ed together is the same as the two variables inverted (complemented) and AND'ed. |
| Second Theorem | The second theorem states that two or more variables NAND'ed together is the same as the two terms inverted (complemented) and OR'ed. |
| Proof | De Morgan's law can be proved mathematically and by using truth tables. |
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What You'll Learn
De Morgan's Law in set theory
De Morgan's laws are a set of rules or laws developed from Boolean expressions for AND, OR, and NOT, using two input variables, A and B. These laws give the relation between union, intersection, and complements in set theory. In other words, they define the relationship between the union, intersection, and complements of sets, and are given for both the complement of the union and the intersection of two sets.
In set theory, De Morgan's laws can be stated as "union and intersection interchange under complementation". This can be expressed as:
$$
{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}}.\end{aligned}}}
$$
De Morgan's first law states that \"The complement of the union of two sets is equal to the intersection of the complements of each set." Let A and B be two sets, then the first law can be expressed mathematically as:
$$
A \cup B)' = A' \cap B'
$$
De Morgan's laws also work for unions and intersections of more than two sets. In Boolean algebra, De Morgan's laws are used to find the equivalency of the NAND and NOR gates.
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Boolean algebra and De Morgan's Law
De Morgan's laws are a set of rules that are used in Boolean algebra and set theory to relate the intersection and union of sets through complements. These laws are useful in simplifying complicated expressions or operations involving "AND", "OR", and "NOT".
In Boolean algebra, De Morgan's laws give the relation between "AND", "OR", and the complements of variables. The first De Morgan's law states that the complement of the OR of two or more variables is equal to the AND of the complement of each variable. This can be expressed as:
$$\overline{A + B} = \overline{A} \cdot \overline{B}$$
The second De Morgan's law states that the complement of the AND of two or more variables is equal to the OR of the complement of each variable. This can be expressed as:
$$\overline{A \cdot B} = \overline{A} + \overline{B}$$
These laws can be used to simplify expressions and are widely used in engineering industries to create hardware and simplify operations. For example, in electronic engineering, De Morgan's laws can be used to develop logic gates using only NAND (AND negated) or NOR (OR negated) gates, resulting in cheaper hardware.
De Morgan's laws can also be applied to text searching using Boolean operators "AND", "OR", and "NOT". For instance, consider a set of documents containing the words "cats" and "dogs". De Morgan's laws state that the searches "(cats OR dogs)" and "NOT (cats AND dogs)" will return the same set of documents.
In summary, De Morgan's laws are a set of rules that are useful in Boolean algebra and set theory for simplifying expressions and understanding relationships between sets. They have applications in various fields, including engineering and computer science.
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De Morgan's Law in computer programming
De Morgan's Law is a fundamental concept in computer programming, providing a way to optimise code and simplify complex logical conditions. It is derived from set theory and Boolean algebra, where it defines the relationship between the union, intersection, and complements of sets. In programming, De Morgan's Law is used to rewrite compound Boolean expressions and negate multiple boolean conditions.
De Morgan's Law consists of two theorems that govern the behaviour of logical operations. The first theorem states that the NOR operation is equivalent to a negative-AND function, while the second theorem asserts that the NAND operation is equivalent to a negative-OR function. These theorems allow for the conversion of one form of a Boolean function into its opposite form by negating the input variables.
In programming, De Morgan's Law is applied to Boolean operators AND, OR, and NOT. For example, consider a set of documents containing the words "cats" and "dogs". De Morgan's Law states that searching for "(cats OR dogs)" will yield the same results as searching for "NOT cats AND NOT dogs". This law can be used to optimise code by reducing the number of logic gates required to perform a particular operation.
Additionally, De Morgan's Law is useful in digital circuit design, where it helps manipulate logic gates. It also aids in finding the conjunctive normal form and disjunctive normal form of a formula in formal logic. While some consider De Morgan's Laws trivial, they are nonetheless valuable in programming. Having a named transformation makes it easier to recognise and confidently apply these laws when needed.
De Morgan's Law is a powerful tool for programmers, enabling them to write more efficient code and better understand the relationship between logical operations. It is a critical concept for any student of programming, providing a foundation for more advanced topics in computer science and logic.
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De Morgan's Law and text searching
De Morgan's laws are a set of two rules or theorems derived from Boolean expressions for AND, OR, and NOT, using two input variables. They allow the conversion of input variables from one form of a Boolean function to another opposite form. These laws are named after Augustus De Morgan, a 19th-century British mathematician.
De Morgan's laws are commonly applied to text searching using Boolean operators AND, OR, and NOT. Consider a set of documents containing the words "cats" and "dogs". De Morgan's laws state that these two searches will return the same set of documents:
- Document 1: Contains only the word "cats".
- Document 2: Contains only "dogs".
- Document 3: Contains both "cats" and "dogs".
- Document 4: Contains neither "cats" nor "dogs".
Search A: (cats OR dogs) will return Documents 1, 2, and 3. So the negation of that search (NOT (cats OR dogs)) will return Document 4.
Search B: (NOT cats) AND (NOT dogs) will return Documents 2 and 4. Applying the AND operator to these two searches will return Document 4.
Thus, Search A and Search B will both return the same set of documents, i.e., Document 4.
In set theory, De Morgan's laws relate the union, intersection, and complements of sets. The first law states that the complement of the union of two sets is equal to the intersection of the complements of each set. For example, if Set A = {1, 3, 5, 7} and Set B = {1, 2, 3}, then the union of A and B is {1, 2, 3, 5, 7}. The complement of this union is {4}, which is the intersection of the complements of Set A and Set B.
In Boolean algebra, De Morgan's laws give the relationship between AND, OR, and the complements of variables. The laws state that the negation of "A or B" is the same as "not A and not B". This can be applied to multiple variables as well.
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De Morgan's Law in circuit diagrams
De Morgan's laws are widely used in electronic engineering to design logic gates and digital circuits. They are also used to develop digital circuits using only NAND or NOR gates, which are often more cost-effective and simpler to fabricate.
De Morgan's laws are derived from Boolean expressions for AND, OR, and NOT using two input variables, A and B. These laws allow the input variables to be negated and converted from one form of a Boolean function into its opposite form. De Morgan's first theorem states that two or more variables NOR'ed together is the same as the two variables inverted (complemented) and AND'ed. The second theorem states that two or more variables NAND'ed together are the same as the two terms inverted (complemented) and OR'ed.
De Morgan's laws can be applied to circuit diagrams by using De Morgan symbols, which are a graphical representation of De Morgan's theorem. These symbols are used to get inversion bubbles to pair up and cancel, and to match a bubble with a signal name that is negated. For example, in a circuit with two AND operations and an OR operation, the schematic can be redrawn with the De Morgan NAND symbol for the third gate. This results in bubbles appearing on both ends of the intermediate nodes, which can be ignored, simplifying the circuit diagram.
De Morgan's laws can also be used to optimise various boolean circuits involving logic gates, allowing the same operation to be performed with fewer apparatus. For example, a standard AND gate function with inverters (NOT gates) on each of its inputs produces an equivalent output condition to a standard NOR gate function. This allows for the simplification of circuit diagrams by reducing the number of gates required.
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Frequently asked questions
De Morgan's Law is a set of two rules or laws that are used in set theory and Boolean algebra. The laws state that the complement of the union of two sets is equal to the intersection of their individual complements, and vice versa.
De Morgan's Law can be used to prove that these two searches will return the same set of documents:
- Search A: Documents containing the word "cats" OR "dogs"
- Search B: Documents containing NOT "cats" AND NOT "dogs"
The first De Morgan's Law is given by the formula: (A ∪ B)’ = A’ ∩ B’ or \(\overline{A + B}\) = \(\overline{A}\)•\(\overline{B}\). The second law is written as: (A ∩ B)’ = A’ ∪ B’ or \(\overline{A.B}\) = \(\overline{A}\) + \(\overline{B}\).
