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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.
Usage on de.wikipedia.org De-morgansche Gesetze; Usage on de.wikibooks.org Digitale Schaltungstechnik/ Schaltalgebra/ De Morgan; Usage on es.wikipedia.org Leyes de De Morgan; Usage on fr.wikibooks.org Électronique numérique : logique/Fonctions logiques élémentaires; Usage on hi.wikipedia.org डिमॉर्गन नियम
A CMOS transistor NAND element. V dd denotes positive voltage.. In CMOS logic, if both of the A and B inputs are high, then both the NMOS transistors (bottom half of the diagram) will conduct, neither of the PMOS transistors (top half) will conduct, and a conductive path will be established between the output and Vss (ground), bringing the output low.
The second De Morgan's law, (¬x) ∨ (¬y) = ¬(x ∧ y), works the same way with the two diagrams interchanged. The first complement law, x ∧ ¬x = 0, says that the interior and exterior of the x circle have no overlap. The second complement law, x ∨ ¬x = 1, says that everything is either inside or outside the x circle.
Diagram of the NAND gates in a CMOS type 4011 integrated circuit. NAND gates are basic logic gates, and as such they are recognised in TTL and CMOS ICs. The standard, 4000 series, CMOS IC is the 4011, which includes four independent, two-input, NAND gates. These devices are available from many semiconductor manufacturers.
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).
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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.