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The nines' complement of a decimal digit is the number that must be added to it to produce 9; the nines' complement of 3 is 6, the nines' complement of 7 is 2, and so on, see table. To form the nines' complement of a larger number, each digit is replaced by its nines' complement.
Two's complement is the most common method of representing signed (positive, negative, and zero) integers on computers, [1] and more generally, fixed point binary values. Two's complement uses the binary digit with the greatest value as the sign to indicate whether the binary number is positive or negative; when the most significant bit is 1 the number is signed as negative and when the most ...
In set theory, the complement of a set A, often denoted by (or A′), [1] is the set of elements not in A. [ 2 ] When all elements in the universe , i.e. all elements under consideration, are considered to be members of a given set U , the absolute complement of A is the set of elements in U that are not in A .
The ones' complement of a binary number is the value obtained by inverting (flipping) all the bits in the binary representation of the number. The name "ones' complement" [1] refers to the fact that such an inverted value, if added to the original, would always produce an "all ones" number (the term "complement" refers to such pairs of mutually additive inverse numbers, here in respect to a ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The same fact can be stated as the indicator function (denoted here by ) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: () = or using the Iverson bracket notation [] = [] [].
Take the complements of each side (in S), using the fact that the complement of a complement of a set is the original set, to obtain X 1 = X 2. This shows that f is one-to-one. Now take any n−k-element subset of S in B, say Y. Its complement in S, Y c, is a k-element subset, and so, an element of A.
What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is 0 when the function R is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY, [2]: 228 the bounded-[2]: 228 and unbounded-[2]: 279 ff mu operators and the CASE function.