Complement

In mathematics the complement of a set X, sometimes written X' or X^c, 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 A^c and B^c be the complements of A and B in N respectively, and define a_c(x) and b_c(x) similar to a(x) and b(x) for A^c and B^c respectively.

Then, a(x)>=b(x) for all x iff b_c(x)>=a_c(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 A^c, so a_c(x)=k(x)-a(x). Similarly, b_c(x)=k(x)-b(x). Solving for k(x),
k(x)=a(x)+a_c(x)=b(x)+b_c(x).
This means that a(x)-b(x)=b_c(x)-a_c(x). Therefore, a(x)>=b(x) for all x iff b_c(x)>=a_c(x) for all x.

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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 arc ∨ angle 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.

 

© Webster 1913.

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

1.

To supply a lack; to supplement.

[R.]

2.

To compliment.

[Obs.]

Jer. Taylor.

 

© Webster 1913.