Search results
Results from the WOW.Com Content Network
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.
A De Morgan symbol can show more clearly a gate's primary logical purpose and the polarity of its nodes that are considered in the "signaled" (active, on) state. Consider the simplified case where a two-input NAND gate is used to drive a motor when either of its inputs are brought low by a switch.
The result is the same as if we shaded that region which is both outside the x circle and outside the y circle, i.e. the conjunction of their exteriors, which is what the left hand side of the law describes. The second De Morgan's law, (¬x) ∨ (¬y) = ¬(x ∧ y), works the same way with the two diagrams interchanged.
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics which work primarily with analog signals. Despite the name, digital electronics designs includes important analog design considerations.
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.
The De Morgan dual is the canonical conjunctive normal form , maxterm canonical form, or Product of Sums (PoS or POS) which is a conjunction (AND) of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.
The principle of inclusion–exclusion, combined with De Morgan's law, can be used to count the cardinality of the intersection of sets as well. Let A k ¯ {\displaystyle {\overline {A_{k}}}} represent the complement of A k with respect to some universal set A such that A k ⊆ A {\displaystyle A_{k}\subseteq A} for each k .
In digital electronics, a NAND gate (NOT-AND) is a logic gate which produces an output which is false only if all its inputs are true; thus its output is complement to that of an AND gate. A LOW (0) output results only if all the inputs to the gate are HIGH (1); if any input is LOW (0), a HIGH (1) output results.