Gavin Howe
De Morgan's laws are a pair of transformation rules in Boolean algebra and set theory that can be used to simplify mathematical expressions. These laws are used to relate the intersection and union of sets through complements. De Morgan's laws can be applied to simplify the denial of $P\implies Q$ as $\lnot (P\implies Q) \iff P\land \lnot Q$. This means that it is not the case that $P$ implies $Q$ if and only if $P$ is true and $Q$ is false. De Morgan's laws can also be used to simplify negations of the some form and the all form, where the negations themselves take on the reversed forms.
| Characteristics | Values |
|---|---|
| Use | Simplifying mathematical expressions |
| Use Case | Boolean algebra and set theory |
| Function | Relating the intersection and union of sets through complements |
| Application | Developing logic gates |
| Application | Creating hardware and simplifying operations |
| Application | Electronics engineering |
| Application | Elementary algebra |
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What You'll Learn
De Morgan's Law of Union
De Morgan's Law, developed by the 19th-century English mathematician Augustus De Morgan, is a pair of transformation rules used in Boolean algebra and set theory. De Morgan's Law of Union is one of the two primary conditions of De Morgan's Law.
The complement of the union of two sets is equal to the intersection of their individual complements. This can be expressed as:
(A ∪ B)’ = A’ ∩ B’
Where:
- A and B are two sets that are subsets of the universal set U
- A' is the complement of A
- B' is the complement of B
- ∪ is used to denote the union
- ∩ is the symbol for intersection
For example, let's say we have two sets, A and B. A' represents the complement of set A, and B' represents the complement of set B. De Morgan's Law of Union states that the complement of the union of sets A and B (A ∪ B)' is equal to the intersection of the complements of A and B (A' ∩ B').
Generalization of De Morgan's Law of Union
The De Morgan's Law of Union can be generalized to more than two sets. If we have n sets given by (A1, A2, ..., An), then the formula is:
(\bigcup_{i = 1}^{n}A_{i})^{'} = \bigcap_{i = 1}^{n} A_{i^{'}}
This means that the complement of the union of n sets is equal to the intersection of the complements of each set.
Applications of De Morgan's Law
De Morgan's Law is widely used in various fields, including mathematics, engineering, and electronics. It is particularly useful for simplifying complex expressions and calculations. In engineering, De Morgan's Law is applied in hardware development and simplifying operations, resulting in cheaper hardware. In electronics, De Morgan's Law is used in developing logic gates, specifically NAND (AND negated) and NOR (OR negated) gates, which are easier to implement in practical circuits.
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De Morgan's Law of Intersection
De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. De Morgan's formulation was influenced by the algebraization of logic undertaken by George Boole.
De Morgan's laws can be used to simplify negations of the "some" form and the "all" form; the negations themselves turn out to have the same forms, but "reversed". That is, the negation of an "all" form is a "some" form, and vice versa. For example, the denial of the sentence "all lawn mowers run on gasoline" is the sentence "some lawn mower does not run on gasoline".
De Morgan's laws can be used in Boolean algebra and set theory to simplify mathematical expressions. In set theory, these laws relate the intersection and union of sets through complements. De Morgan's first law states that the complement of the union of two sets A and B is equal to the intersection of the complement of the sets A and B. This can be represented as (A ∪ B)’ = A’ ∩ B’.
De Morgan's laws can also be used to simplify the denial of $P\implies Q$. This can be represented as:
\eqalign{
\lnot (P\implies Q) & \iff \lnot (\lnot P\lor Q)\cr
& \iff (\lnot\lnot P)\land (\lnot Q)\cr
& \iff P\land \lnot Q\cr
}
So, the denial of $P\implies Q$ is $P\land \lnot Q$. In other words, it is not the case that $P$ implies $Q$ if and only if $P$ is true and $Q$ is false.
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Denials of formulas
De Morgan's laws can be used to simplify the denial of a formula. A formula is usually simpler if the negation symbol ($\lnot$) does not appear in front of any compound expression. Instead, it should appear only in front of simple statements such as $P(x)$.
$$
\eqalign{
\lnot \forall x (P(x)\lor \lnot Q(x))&\iff \exists x \lnot(P(x)\lor \lnot Q(x))\cr
&\iff\exists x (\lnot P(x)\land \lnot \lnot Q(x))\cr
&\iff\exists x (\lnot P(x)\land Q(x)) \cr
}
$$
This can also be applied to the denial of $P \implies Q$, which can be simplified using De Morgan's laws:
$$
\eqalign{
\lnot (P\implies Q) & \iff \lnot (\lnot P\lor Q)\cr
& \iff (\lnot\lnot P)\land (\lnot Q)\cr
& \iff P\land \lnot Q\cr
}
$$
So, the denial of $P \implies Q$ is $P \land \lnot Q$. In other words, it is not the case that $P$ implies $Q$ if and only if $P$ is true and $Q$ is false.
De Morgan's laws can be used to simplify negations of the "some" form and the "all" form. The negations themselves turn out to have the same forms, but "reversed". That is, the negation of an "all" form is a "some" form, and vice versa. For example, the denial of the sentence "all lawn mowers run on gasoline" is the sentence "some lawn mower does not run on gasoline".
It is easy to confuse the denial of a sentence with something stronger. If the universe is the set of all people, the denial of the sentence "All people are tall" is not the sentence "No people are tall". This might be called the opposite of the original sentence—it says more than simply "'All people are tall' is untrue". The correct denial of this sentence is "there is someone who is not tall", which is a considerably weaker statement.
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Boolean algebra
DeMorgan's Law, also referred to as De Morgan's Laws or DeMorgan's Theorems, is a critical concept in Boolean algebra and set theory. It is used to simplify mathematical expressions and establish a relationship between union and intersection via complementation.
DeMorgan's Law of Union states that the complement of the union of two sets is equal to the intersection of their individual complements. In other words, if we have two sets, A and B, the complement of their union (A ∪ B)' is equal to the intersection of the complement of A and the complement of B (A' ∩ B'). This can be represented as: (A ∪ B)’ = A’ ∩ B’.
DeMorgan's Law of Intersection states that the complement of the intersection of two sets is equal to the union of their individual complements. Using the same sets A and B, the complement of their intersection (A ∩ B)' is equal to the union of the complement of A and the complement of B (A' ∪ B'). This can be represented as: (A ∩ B)’ = A’ ∪ B’.
These laws are not limited to just two sets and can be generalized to multiple sets. For example, if we have n sets given by (A1, A2, ..., An), the formula for De Morgan's Law of Union becomes: (∪i = 1Ai)′ = ∩i = 1Ai′.
DeMorgan's Theorems are particularly useful in electronics engineering for developing logic gates. They describe the equivalence between gates with inverted inputs and gates with inverted outputs. For instance, DeMorgan's First Theorem proves that when two or more input variables are AND'ed and negated, they are equivalent to the OR of the complements of the individual variables. This results in a NAND gate being equivalent to a Negative-OR gate. Similarly, DeMorgan's Second Theorem proves that when two or more input variables are OR'ed and negated, they are equivalent to the AND of the complements of the individual variables, resulting in a NOR gate being equivalent to a Negative-AND gate.
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Set theory
De Morgan's laws are a pair of transformation rules that can be applied to both set theory and boolean algebra. They are used to relate the intersection and union of sets through complements. In set theory, De Morgan's laws relate the three fundamental set operations: the set union, set intersection, and set complement.
De Morgan's first law states that the complement of the union of two sets is the intersection of their complements. This can be expressed as: (AUB)’ = A’∩B’. In other words, the complement of the union of any two sets, A and B, is equal to the intersection of their complements.
De Morgan's second law, also known as the law of intersection, states that the complement of the intersection of two sets is the union of their complements. This can be expressed as: (A ∩ B)’ = A’ ∪ B’. In other words, the complement of the intersection of any two sets, A and B, is equal to the union of their complements.
These laws can be visualised using Venn diagrams and can be used to simplify mathematical expressions. They are also used in computer engineering for developing logic gates and in probability theory.
De Morgan's laws can also be used to simplify the denial of $P\implies Q$. This can be expressed as:
> $\lnot (P\implies Q) \iff \lnot (\lnot P\lor Q)$
> $\iff (\lnot\lnot P)\land (\lnot Q)$
> $\iff P\land \lnot Q$
So, the denial of $P\implies Q$ is $P\land \lnot Q$. In other words, it is not the case that $P$ implies $Q$ if and only if $P$ is true and $Q$ is false.
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