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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.
Integrated injection logic (IIL, I 2 L, or I2L) is a class of digital circuits built with multiple collector bipolar junction transistors (BJT). [1] When introduced it had speed comparable to TTL yet was almost as low power as CMOS, making it ideal for use in VLSI (and larger) integrated circuits. The gates can be made smaller with this logic ...
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
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.
This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples. More generally, ∧ (¬ x i) = ¬ ∨ x i. The left side is true if and only if ∀i.¬x i, and the right side if and only if ¬∃i.x i.
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.
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.
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).