Home Proof of De Morgan’s Law with Logic
Technology

Proof of De Morgan’s Law with Logic

About De Morgan’s Law

De Morgon’s Law states that the complement of the union of 2 sets is that the intersection of their enhances and also the complement of the intersection of 2 sets is that the union of their enhances. These square measure mentioned once the goodman of science Diamond State Morgan. This law may be expressed as ( A ∪ B) ‘ = A ‘ ∩ B ‘. In pure mathematics, these laws relate to the intersection and union of sets by enhances.

De Morgan’s Law: Logic and Proof

De Morgan’s Laws Statement and Proof

A well-defined assortment of objects or components is understood as a collection. numerous operations like complement of a collection, union, and intersection may be performed on 2 sets. These operations and their usage may be additionally simplified employing a set of laws called Diamond State Morgan’s Laws. These square measure simple and straightforward laws.

Any set consisting of all the objects or components associated with a selected context is outlined as a universal set. take into account a universal set U such A and B square measure the subsets of this universal set.

According to Diamond State Morgan’s 1st law, the complement of the union of 2 sets A and B are capable of the intersection of the complement of the sets A and B.

(A∪B)’= A’∩ B’      —–(1)

Where complement of a collection is outlined as

A’= Where A’ denotes the complement.

This law may be simply envisioned victimization logician Diagrams.

For any 2 finite sets A and B;

(i) (A U B)’ = A’ ∩ B’ (which could be a Diamond State Morgan’s law of union).

(ii) (A ∩ B)’ = A’ U B’ (which could be a Diamond State Morgan’s law of intersection).

Proof of Diamond State Morgan’s law:

(A ∩ B)’ = A’ U B’

Let M = (A ∩ B)’ and N = A’ U B’

Let x be an associate discretionary component of M then x ∈ M ⇒ x ∈ (A ∩ B)’

⇒ x ∉ (A ∩ B)

⇒ x ∉ A or x ∉ B

⇒ x ∈ A’ or x ∈ B’

⇒ x ∈ A’ U B’

⇒ x ∈ N

Therefore, M ⊂ N …………….. (i)

Again, let y be an associate discretionary component of N then y ∈ N ⇒ y ∈ A’ U B’

⇒ y ∈ A’ or y ∈ B’

⇒ y ∉ A or y ∉ B

⇒ y ∉ (A ∩ B)

⇒ y ∈ (A ∩ B)’

⇒ y ∈ M

Therefore, N ⊂ M …………….. (ii)

Now mix (i) and (ii) we tend to get; M = N i.e. (A ∩ B)’ = A’ U B’

Let P = (A U B)’ and Q = A’ ∩ B’

Let x be an associate discretionary component of P then x ∈ P ⇒ x ∈ (A U B)’

⇒ x ∉ (A U B)

⇒ x ∉ A and x ∉ B

⇒ x ∈ A’ and x ∈ B’

⇒ x ∈ A’ ∩ B’

⇒ x ∈ Q

Therefore, P ⊂ Q …………….. (i)

Again, let y be an associate discretionary component of Q then y ∈ Q ⇒ y ∈ A’ ∩ B’

⇒ y ∈ A’ and y ∈ B’

⇒ y ∉ A and y ∉ B

⇒ y ∉ (A U B)

⇒ y ∈ (A U B)’

⇒ y ∈ P

Therefore, Q Q P …………….. (ii)

Now mix (i) and (ii) we tend to get; P = Q i.e. (A U B)’ = A’ ∩ B’

De Morgan’s Law: Logic and Proof

Demorgans law java

Some of a lot of common iterations for Diamond State Morgan’s Law:

!(A && B) is that the same as!A || !B

!(A || B) is that the same as!A && !B

!(C > D) is that the same as C <= D

!(C < D) is that the same as C >= D

!(C >= D) is that the same as C < D

!(C <= D) is that the same as C > D

!(E == F) is that the same as E != F

!(E != F) is that the same as E == F

Remember, you are doing not ought to limit yourself to 2 operands after you work with the “AND” operator and also the “OR” operator. !(A && B && C) is that the same as !A||!B||!C. For the iterations shown higher than, A and B ought to be Boolean values, C and D ought to be numbers and E and F are also a spread of information varieties.

De Morgan’s Laws describe however mathematical statements and ideas square measure connected through their opposites. In pure mathematics, Diamond State Morgan’s Laws relate the intersection and union of sets through enhances. In symbolic logic, Diamond State Morgan’s Laws relate conjunctions and disjunctions of propositions through negation. Diamond State Morgan’s Laws also are applicable in laptop engineering for developing logic gates.

De Morgan’s Law: Logic and Proof

The rules may be expressed in English as:

the negation of a disjunction is that the conjunction of the negations; and

the negation of a conjunction is that the disjunction of the negations;

or

the complement of the union of 2 sets is that the same because the intersection of their complements; and

the complement of the intersection of 2 sets is the same because the union of their enhances.

or

not (A or B) = not A and not B; and

not (A and B) = not A or not B