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In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
A logic gate is a device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has, for instance, zero rise time and unlimited fan-out, or it may refer to a non-ideal physical device [1] (see ...
Alternatively, in untyped logic, we can require to be false whenever is an ur-element. In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set. To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:
Affine logic, which forbids contraction but allows global weakening (a decidable extension). Strict logic or relevant logic, which forbids weakening but allows global contraction. Non-commutative logic or ordered logic, which removes the rule of exchange, in addition to barring weakening and contraction. In ordered logic, linear implication ...
The 3-input Fredkin gate is functionally complete reversible gate by itself – a sole sufficient operator. There are many other three-input universal logic gates, such as the Toffoli gate. In quantum computing, the Hadamard gate and the T gate are universal, albeit with a slightly more restrictive definition than that of functional completeness.
In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table:
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original.