What is "complement"

Complement \Com"ple*ment\, v. t.

1. To supply a lack; to supplement. [R.]

2. To compliment. [Obs.]
   --Jer. Taylor.

Complement \Com"ple*ment\, n. [L. complementun: cf. F. compl['e]ment. See Complete, v. t., and cf. Compliment.]

1. That which fills up or completes; the quantity or number required to fill a thing or make it complete.

2. That which is required to supply a deficiency, or to complete a symmetrical whole.

   History is the complement of poetry.
   --Sir J. Stephen.

3. Full quantity, number, or amount; a complete set; completeness.

   To exceed his complement and number appointed him which was one hundred and twenty persons.
   --Hakluyt.

4. (Math.) A second quantity added to a given quantity to make it equal to a third given quantity.

5. Something added for ornamentation; an accessory. [Obs.]

   Without vain art or curious complements.
   --Spenser.

6. (Naut.) The whole working force of a vessel.

7. (Mus.) The interval wanting to complete the octave; -- the fourth is the complement of the fifth, the sixth of the third.

8. A compliment. [Obs.]
   --Shak.

   Arithmetical compliment of a logarithm. See under Logarithm.

   Arithmetical complement of a number (Math.), the difference between that number and the next higher power of 10; as, 4 is the complement of 6, and 16 of 84.

   Complement of an arc or Complement of an angle (Geom.), the difference between that arc or angle and 90[deg].

   Complement of a parallelogram. (Math.) See Gnomon.

   In her complement (Her.), said of the moon when represented as full.

late 14c., "that which completes," from Old French compliement "accomplishment, fulfillment" (14c., Modern French complément), from Latin complementum "that which fills up or completes," from complere "fill up" (see complete (adj.)). Originally also having senses which were taken up c.1650-1725 by compliment.

1610s, "exchange courtesies," from complement (n.). Meaning "make complete" is from 1640s. Related: Complemented; complementing.

n. 1 (context now rare English) Something (or someone) that completes; the consummation. (from 14th c.) 2 (context obsolete English) The act of completing something, or the fact of being complete; completion, completeness, fulfilment. (15th-18th c.) 3 The totality, the full amount or number which completes something. (from 16th c.) 4 (context obsolete English) Something which completes one's equipment, dress etc.; an accessory. (16th-17th c.) 5 (context nautical English) The whole working force of a vessel. 6 (context heraldry English) fullness (of the moon). (from 17th c.) 7 (context astronomy geometry English) An angle which, together with a given angle, makes a right angle. (from 18th c.) 8 Something which completes, something which combines with something else to make up a complete whole; loosely, something perceived to be a harmonious or desirable partner or addition. (from 19th c.) 9 (context grammar English) A word or group of words that completes a grammatical construction in the predicate and that describes or is identified with the subject or object. (from 19th c.) 10 (context music English) An interval which, together with the given interval, makes an octave. (from 19th c.) 11 (context optics English) The color which, when mixed with the given color, gives black (for mixing pigments) or white (for mixing light). (from 19th c.) 12 (context set theory English) Given two sets, the set containing one set's elements that are not members of the other set (whether a relative complement or an absolute complement). (from 20th c.) 13 (context immunology English) One of several blood proteins that work with antibody during an immune response. (from 20th c.) 14 (context logic English) An expression related to some other expression such that it is true under the same conditions that make other false, and vice vers

1. (from 20th c.) 15 (context electronics English) A voltage level with the opposite logical sense to the given one. 16 (context computing English) A bit with the opposite value to the given one; the logical complement of a number. 17 (context computing mathematics English) The diminished radix complement of a number; the nines' complement of a decimal number; the ones' complement of a binary number. 18 (context computing mathematics English) The radix complement of a number; the two's complement of a binary number. 19 (context computing mathematics English) The numeric complement of a number. 20 (context genetics English) A nucleotide sequence in which each base is replaced by the complementary base of the given sequence: adenine (A) by thymine (T) or uracil (U), cytosine (C) by guanine (G), and vice versa. 21 (obsolete spelling of compliment English) v

2. 1 To complete, to bring to perfection, to make whole. 2 To provide what the partner lacks and lack what the partner provides. 3 To change a voltage, number, color, et

3. to its complement. 4 (obsolete form of compliment English)

1. n. a word or phrase used to complete a grammatical construction

2. a complete number or quantity; "a full complement"

3. number needed to make up whole force; "a full complement of workers" [syn: full complement]

4. something added to complete or make perfect; "a fine wine is a perfect complement to the dinner"

5. one of a series of enzymes in the blood serum that are part of the immune response

6. either of two parts that mutually complete each other

7. v. make complete or perfect; supply what is wanting or form the complement to; "I need some pepper to complement the sweet touch in the soup"

In set theory, the complement of a set refers to elements not in . The relative complement of with respect to a set is the set of elements in but not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is the set of elements in but not in .

In grammar, a complement is a word, phrase or clause that is necessary to complete the meaning of a given expression. Complements are often also arguments (i.e., expressions that help complete the meaning of a predicate).

Complement may refer to:

• Complement (linguistics), a word or phrase having a particular syntactic role
• Phonetic complement
• Complementary, a type of opposite in lexical semantics (sometimes called an antonym)
• Complement (music), an interval that when added to another spans an octave
• Aggregate complementation (music), the separation of pitch-class collections into complementary sets
• Complementary color, in painting and optics
• Complement good (economics), a good often consumed together with another good
• Ship's complement, the number of persons in a ship's company, including both commissioned officers and crew

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.

In interval complementation a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its inverse or inversion. Note that the octave and the unison are each other's complements and that the tritone is its own complement (though the latter is "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context).

In the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale contains all the other notes of the scale. For example, A-B-C-D-E-F-G is complemented by B-C-E-F-A.

Note that musical set theory broadens the definition of both senses somewhat.

In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem.

For example, one important problem is whether a number is a prime number. Its complement is to determine whether a number is a composite number (a number which is not prime). Here the domain of the complement is the set of all integers exceeding one.

There is a Turing reduction from every problem to its complement problem. The complement operation is an involution, meaning it "undoes itself", or the complement of the complement is the original problem.

One can generalize this to the complement of a complexity class, called the complement class, which is the set of complements of every problem in the class. If a class is called C, its complement is conventionally labelled co-C. Notice that this is not the complement of the complexity class itself as a set of problems, which would contain a great deal more problems.

A class is said to be closed under complement if the complement of any problem in the class is still in the class. Because there are Turing reductions from every problem to its complement, any class which is closed under Turing reductions is closed under complement. Any class which is closed under complement is equal to its complement class. However, under many-one reductions, many important classes, especially NP, are believed to be distinct from their complement classes (although this has not been proven).

The closure of any complexity class under Turing reductions is a superset of that class which is closed under complement. The closure under complement is the smallest such class. If a class is intersected with its complement, we obtain a (possibly empty) subset which is closed under complement.

Every deterministic complexity class (DSPACE(f(n)), DTIME(f(n)) for all f(n)) is closed under complement, because one can simply add a last step to the algorithm which reverses the answer. This doesn't work for nondeterministic complexity classes, because if there exist both computation paths which accept and paths which reject, and all the paths reverse their answer, there will still be paths which accept and paths which reject — consequently, the machine accepts in both cases.

Some of the most surprising complexity results shown to date showed that the complexity classes NL and SL are in fact closed under complement, whereas before it was widely believed they were not (see Immerman–Szelepcsényi theorem). The latter has become less surprising now that we know SL equals L, which is a deterministic class.

Every class which is low for itself is closed under complement.

In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that

G = HK = {hk : h ∈ H, k ∈ K}and H ∩ K = {e}.
Equivalently, every element of G has a unique expression as a product hk where h ∈ H and k ∈ K. This relation is symmetrical: if K is a complement of H, then H is a complement of K. Neither H nor K need be a normal subgroup of G.

Complements generalize both the direct product (where the subgroups H and K commute element-wise), and the semidirect product (where one of H or K normalizes the other). The product corresponding to a general complement is called the Zappa–Szép product. In all cases, complement subgroups factor a group into smaller pieces.

Some properties of complement subgroups:

• Complements need not exist, and if they do they need not be unique. That is, H could have two distinct complements K and K in G.
• If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group.
• If K is a complement of H in G then K forms both a left and right transversal of H (that is, the elements of K form a complete set of representatives of both the left and right cosets of H).
• the Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups.

A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with p-complements for every prime p; these p-complements are used to form what is called a Sylow system.

A Frobenius complement is a special type of complement in a Frobenius group.

A complemented group is one where every subgroup has a complement.