Sheffer stroke
Sheffer stroke, also known as the nand operation, is a logical operation on two logical values, typically the values of two propositions, that produces a value of false only if both of its operands are true. In other words, it acts as the negation of the AND operation. The operation is named after Henry M. Sheffer, who in 1913 identified that all logical operations could be expressed in terms of a single, fundamental operation such as the nand operation. This discovery has significant implications in the fields of logic, mathematics, and computer science, particularly in the design and operation of digital circuits.
Definition
The Sheffer stroke is denoted by the symbol '|', and its operation can be defined by the truth table:
| p | q | q |
|---|---|---|
| T | T | F |
| T | Q | T |
| Q | T | T |
| Q | Q | T |
Where T stands for true and F for false. The result of p | q is true except when both p and q are true.
Properties
The Sheffer stroke possesses several important properties that make it a fundamental operation in logic:
- Functionally complete: Any logical operation can be constructed using only the Sheffer stroke. This means that logical expressions and functions can be simplified to expressions involving only nand operations.
- Self-dual: The Sheffer stroke operation is self-dual, meaning that swapping all Ts for Fs and all Fs for Ts in its truth table, and then swapping the operands, results in the same operation.
- Associative: The operation is associative, a property that allows the grouping of operands without affecting the result.
Applications
The Sheffer stroke's property of functional completeness makes it invaluable in the design of digital circuits and computational logic. It is used in the construction of logic gates and the simplification of logical expressions, which are fundamental in the development of digital computers and other digital systems.
See also
- Logical conjunction
- Logical disjunction
- Logical negation
- Boolean algebra (logic)
- Functional completeness
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