Logical Nand

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Information About Logical Nand

	CATEGORIES ABOUT SHEFFER STROKE
	logic
	boolean algebra
	history of logic
	logic gates
	nand gate

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	{ Class	"wikitable"

"''p'' implies ''q''" is equivalent to "(''p'' NAND ''q'') NAND ''p''"
$p\; ightarrow\; q\; \backslash equiv\; \backslash overline\{\backslash overline\{(p\; \backslash cdot\; q)\}\; \backslash cdot\; p\}$



This leads to an alternative axiom system for Boolean Algebra s, requiring but one operation.

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Digital systems employing certain logic circuits take advantage of this property. In complicated logical expressions, normally written in terms of other logic functions such as AND , OR , and NOT , writing these in terms of NAND saves on cost, because implementing such circuits using NAND gate yields a more compact result than the alternatives.

The dual of NAND, the operator NOR, also suffices to express all Boolean operations.


SHEFFER STROKE

  { Class "wikitable"
  Eg A A A, </math>
  :<math> A \wedge B (A B) (A B), </math>
  :<math> A Ee B (A A) (B B), </math>
  :<math> A Ightarrow B A (B B) = A (A B) </math>
  '' "http://wwwinformationdelightinfo/encyclopedia/entry/Frege's_propositional_calculus" class="copylinks">THEN-1 :'' (U(U(V(UU))))
  ''Commutativity:'' (XY) (YX)<br>
  ''Double Negation:'' ((XX)(XX)) X<br>
  ''Mimesis:'' (U(XX)) (U(UX)) <br>
  '' "http://wwwinformationdelightinfo/encyclopedia/entry/Frege's_propositional_calculus" class="copylinks">THEN-3 :'' (U(U(V(VX)))) = (V(V(U(UX))))
  Note The Formula (U(VX)) Has The Interpretation U&rarrV&andX "http://wwwinformationdelightinfo/encyclopedia/entry/Modus_ponens" class="copylinks">Modus Ponens is the special case of MP-1 and MP-2 when V and X are identical