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AND


ANDGate

A connective in logic which yields true if all conditions are true, and false if any condition is false. A AND B is denoted A ^ B (Mendelson 1997, p. 12), A&B, A intersection B (Simpson 1987, p. 538), A·B, A.B (Carnap 1958, p. 7), or simply AB (Simpson 1987, p. 538). The way to distinguish the similar symbols  ^ (AND) and  v (OR) is to note that the symbol for AND is oriented in the same direction as the capital letter 'A.' The AND operation is implemented in the Wolfram Language as And[A, B, ...]. The circuit diagram symbol for an AND gate is illustrated above.

The AND operation ( ^ ) can be written in terms of NOT (!) and OR ( v ) as

 A ^ B=!(!A v !B).

The binary AND operator has the following truth table (Carnap 1958, p. 10; Simpson 1987, p. 545; Mendelson 1997, p. 12).

ABA ^ B
TTT
TFF
FTF
FFF

A product of ANDs (the AND of n conditions) is called a conjunction, and is denoted

  ^ _(k=1)^nA_k.

For example, the truth table for A AND B AND C is given below (Simpson 1987, p. 545).

ABCA ^ B ^ C
TTTT
TTFF
TFTF
TFFF
FTTF
FTFF
FFTF
FFFF
BitAnd

Two binary numbers can have the operation AND performed bitwise with 1 representing true and 0 false. Some computer languages denote this operation on A, B, and C as A && B && C or logand(A,B,C). Bitwise AND is implemented in the Wolfram Language as BitAnd[n1, n2, ...]. The illustration above plots the bitwise AND of the array of numbers from -31 to 31 (Wolfram 2002, p. 871).


See also

Binary Operator, Conjunction, Connective, Intersection, NAND, NOR, NOT, OR, Truth Table, Wedge, XNOR, XOR

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References

Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, pp. 7 and 10, 1958.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 12, 1997.Simpson, R. E. "The AND Gate." §12.5.2 in Introductory Electronics for Scientists and Engineers, 2nd ed. Boston, MA: Allyn and Bacon, pp. 538 and 544-546, 1987.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 871, 2002.

Cite this as:

Weisstein, Eric W. "AND." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AND.html

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