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Negated AND Gate
In digital electronics, a N AND gate
AND gate
(negative-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 the AND gate. A LOW (0) output results only if both the inputs to the gate are HIGH (1); if one or both inputs are LOW (0), a HIGH (1) output results. It is made using transistors and junction diodes. By De Morgan's theorem, AB=A+B, and thus a N AND gate
AND gate
is equivalent to inverters followed by an OR gate. The N AND gate
AND gate
is significant because any boolean function can be implemented by using a combination of NAND gates. This property is called functional completeness. It shares this property with the NOR gate. Digital systems employing certain logic circuits take advantage of NAND's functional completeness. The function NAND(a1, a2, ..., an) is logically equivalent to NOT(a1 AND a2 AND ..
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Digital Electronics
Digital electronics
Digital electronics
or digital (electronic) circuits are electronics that operate on digital signals. In contrast, analog circuits manipulate analog signals whose performance is more subject to manufacturing tolerance, signal attenuation and noise. Digital techniques are helpful because it is a lot easier to get an electronic device to switch into one of a number of known states than to accurately reproduce a continuous range of values. Digital electronic circuits are usually made from large assemblies of logic gates (often printed on integrated circuits), simple electronic representations of Boolean logic
Boolean logic
functions.[1]Contents1 History 2 Properties 3 Construction 4 Design4.1 Structure of digital systems4.1.1 Representation 4.1.2 Combinational vs
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Logical Disjunction
In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +. "A or B" is true if A is true, or if B is true, or if both A and B are true. In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case. An operand of a disjunction is called a disjunct. Related concepts in other fields are:In natural language, the coordinating conjunction "or". In programming languages, the short-circuit or control structure. In set theory, union. In predicate logic, existential quantification.Contents1 Notation 2 Definition2.1 Truth table3 Properties 4 Symbol 5 Applications in computer science5.1 Bitwise operation 5.2 Logical oper
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Xnor Gate
The XNOR gate (sometimes, EXNOR, ENOR, and, rarely[dubious – discuss], NXOR, XAND) is a digital logic gate whose function is the logical complement of the exclusive OR (XOR) gate. The two-input version implements logical equality, behaving according to the truth table to the right. A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results. The algebraic notation used to represent the XNOR operation is S = A ⊙ B displaystyle S=Aodot B .Contents1 Symbols 2 Hardware description and pinout 3 Alternatives 4 See also 5 ReferencesSymbols[edit] There are two symbols for XNOR gates: one with distinctive shape and one with rectangular shape and label
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Boolean Algebra
In mathematics and mathematical logic, Boolean algebra
Boolean algebra
is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively
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Logical Connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective. Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic
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Tautology (logic)
In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation. A simple example is "(x equals y) or (x does not equal y)" (or as a less abstract example, "The ball is green or the ball is not green"). Philosopher
Philosopher
Ludwig Wittgenstein
Ludwig Wittgenstein
first applied the term to redundancies of propositional logic in 1921. (It had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternative sense.) A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent
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Logical Truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.[1] Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. It must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true. A logical truth is considered by some philosophers to be a statement which is true in all possible worlds
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Sheffer Stroke
In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "" (see vertical bar, not to be confused with "" which is often used to represent disjunction), "Dpq", or "↑" (an upwards arrow), denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation. Like its dual, the NOR operator (also known as the Peirce arrow
Peirce arrow
or Quine dagger), NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete)
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Converse Implication
Converse implication
Converse implication
is the converse of implication. That is to say; that for any two propositions P and Q, if Q implies P, then P is the converse implication of Q. It may take the following forms:p⊂q, Bpq, or p←qContents1 Definition1.1 Truth table 1.2 Venn diagram2 Properties 3 Symbol 4 Natural language 5 Boolean Algebra 6 See alsoDefinition[edit] Truth table[edit] The truth table of A⊂BA B ⊂T T TT F TF T FF F TVenn diagram[edit] The Venn diagram
Venn diagram
of "If B then A" (the white area shows where the statement is false)Properties[edit] truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of converse implication. Symbol[edit]This section is empty. You can help by adding to it
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Material Conditional
The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form p → q (termed a conditional statement) which is read as "if p then q". Unlike the English construction "if... then...", the material conditional statement p → q does not specify a causal relationship between p and q
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IMPLY Gate
The IMPLY gate is a digital logic gate that implements a logical conditional.Contents1 Symbols 2 Implementations 3 See also 4 ReferencesSymbols[edit] There are two symbols for IMPLY gates: the traditional symbol and the IEEE symbol
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Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false, and false when p is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa
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NOR Gate
The N OR gate
OR gate
is a digital logic gate that implements logical NOR - it behaves according to the truth table to the right. A HIGH output (1) results if both the inputs to the gate are LOW (0); if one or both input is HIGH (1), a LOW output (0) results. NOR is the result of the negation of the OR operator. It can also be seen as an AND gate
AND gate
with all the inputs inverted. NOR is a functionally complete operation—NOR gates can be combined to generate any other logical function. it shares this property with the NAND gate. By contrast, the OR operator is monotonic as it can only change LOW to HIGH but not vice versa. In most, but not all, circuit implementations, the negation comes for free—including CMOS
CMOS
and TTL. In such logic families, OR is the more complicated operation; it may use a NOR followed by a NOT
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Exclusive Or
but not is Venn diagram
Venn diagram
of A ⊕ B ⊕ C displaystyle scriptstyle Aoplus Boplus C   ⊕   displaystyle ~oplus ~   ⇔   displaystyle ~Leftrightarrow ~ Exclusive or
Exclusive or
or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).[1] It is symbolized by the prefix operator J[2] and by the infix operators XOR (/ˌɛks ˈɔːr/), EOR, EXOR, ⊻, ⩒, ⩛, ⊕, ↮, and ≢. The negation of XOR is logical biconditional, which outputs true only when both inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both"
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Logical Biconditional
In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "p if and only if q", where p is an antecedent and q is a consequent.[1] This is often abbreviated "p iff q". The operator is denoted using a doubleheaded arrow (↔), a prefixed E (Epq), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to (p → q) ∧ (q → p). It is also logically equivalent to (p ∧ q) ∨ (¬p ∧ ¬q) (or the XNOR (exclusive nor) boolean operator), meaning "both or neither". The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false. In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the sets a and b coincide: they are identical
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