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De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
By De Morgan's laws, a two-input NAND gate's logic may be expressed as ¯ ¯ = ¯, making a NAND gate equivalent to inverters followed by an OR gate. The NAND gate is significant because any Boolean function can be implemented by using a combination of NAND gates.
An XNOR gate is made by considering the disjunctive normal form + ¯ ¯, noting from de Morgan's law that a NAND gate is an inverted-input OR gate. This construction entails a propagation delay three times that of a single NAND gate and uses five gates.
PMOS gates have the same arrangement as NMOS gates if all the voltages are reversed. [22] Thus, for active-high logic, De Morgan's laws show that a PMOS NOR gate has the same structure as an NMOS NAND gate and vice versa.
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = ([0, 1], max( x , y ), min( x , y ), 0, 1, 1 − x ) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.
They thus obey the laws of classical propositional logic (such as de Morgan's laws) with the set operations of union and intersection corresponding to the Boolean conjunctives and subset inclusion corresponding to material implication. In fact, a stronger claim is true: they must obey the infinitary logic L ω 1,ω.