In mathematics the complement of a set X, sometimes written X' or Xc, is the set of all the elements in the universe under consideration excluding those in the specified set. So if the current universe is the set of natural numbers and X is the set of even numbers then X' is the set of odd numbers.

In logic the complement of a proposition is the negation of that proposition. For example the complement of "the sun is shining" would be "the sun is not shining". For this reason the unary function in a Boolean algebra is often referred to as the complement operation.

The greater than theorem on some sets and their complements:

Let N be the set of natural numbers. Let A and B be non-empty proper subsets of N such that A is not a subset of B nor is B a subset of A.
Let f(x) be a one-to-one function with domain N and range N where f(x+1)>f(x) for all x. Let R be the set of natural numbers in the range [f(x)+1, f(x+1)]. Let k(x)=f(x+1)-f(x), which is the total number of elements in the range R. Let a(x) be the number of elements in A which are also in R; similarly, let b(x) be the number of elements in B which are also in R.

Let Ac and Bc be the complements of A and B in N respectively, and define ac(x) and bc(x) similar to a(x) and b(x) for Ac and Bc respectively.

Then, a(x)>=b(x) for all x iff bc(x)>=ac(x) for all x.

Proof:

k(x) is the number of elements in R (a finite number since it is a range with definite endpoints in N). 0<=a(x)<=k(x) for all x since a(x) is the number of elements in common between A and R. 0<=b(x)<=k(x) for all x since b(x) is the number of elements in common between B and R.
By the definition of complement set, if an element t is in A, then it is not in Ac, so ac(x)=k(x)-a(x). Similarly, bc(x)=k(x)-b(x). Solving for k(x),
k(x)=a(x)+ac(x)=b(x)+bc(x).
This means that a(x)-b(x)=bc(x)-ac(x). Therefore, a(x)>=b(x) for all x iff bc(x)>=ac(x) for all x.

This theorem is obscure to say the least, but it may prove useful very soon.

Com"ple*ment (?), n. [L. complementum: cf. F. complément. 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 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 arcangle Geom., the difference between that arc or angle and 90°. -- Complement of a parallelogram. Math. See Gnomon. -- In her complement Her., said of the moon when represented as full.

Com"ple*ment (?), v. t.

1.

To supply a lack; to supplement.

[R.]

2.

To compliment.

[Obs.]

Jer. Taylor.

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