De Morgan’s laws
In formal logic, De Morgan's laws are rules relating the logical operators "and" and "or" in terms of each other via negation, namely: (A U B)’=A’ ∩ B’
(A ∩ B)’=A’ U B’
The rules can be expressed in English as:
"The negation of a conjunction is the disjunction of the negations." and "The negation of a disjunction is the conjunction of the negations." The law is named after Augustus De Morgan (1806–1871) who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Although a similar observation was made by Aristotle and was known to Greek and Medieval logicians (in the 14th century William of Ockham wrote down the words that would result by reading the laws out), De Morgan is given credit for stating the laws formally and incorporating them in to the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial. Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments. Logic gates invert all inputs to a gate reverses that gate's essential function from AND to OR, or vice versa, and also inverts the output. So, an OR gate with all inputs inverted (a Negative-OR gate) behaves the same as a NAND gate, and an AND gate with all inputs inverted (a Negative-AND gate) behaves the same as a NOR gate. DeMorgan's theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs

De Morgan’s theorem is used to simplify a lot expression of complicated logic gates. For example, (A + (BC)')'. The parentheses symbol is used in the example.

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The answer is A BC.

Let's apply the principles of DeMorgan's theorems to the simplification of a gate circuit:

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