cardinality of monomials
Theorem 1.
If is a finite set of variable symbols, then the number of monomials
of
degree constructed from these symbols is , where
is the cardinality of .
Proof.
The proof proceeds by inducion on the cardinality of . If has but one
element, then there is but one monomial of degree , namely the sole element
of raised to the -th power. Since , the
conclusion holds when .
Suppose, then, that the result holds whenver for some . Let be
a set with exactly elements and let be an element of . A monomial
of degree constructed from elements of can be expressed as the product
of a power of and a monomial constructed from the elements of . By the induction hypothesis, the number of monomials of degree
constructed from elements of is .
Summing over the possible powers to which may be raised, the number of
monomials of degree constructed from the elements of is as follows:
∎
Theorem 2.
If is an infinite set of variable symbols, then the number of monomials
of degree constructed from these symbols equals the cardinality of .
Title | cardinality of monomials |
---|---|
Canonical name | CardinalityOfMonomials |
Date of creation | 2013-03-22 16:34:42 |
Last modified on | 2013-03-22 16:34:42 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 10 |
Author | rspuzio (6075) |
Entry type | Theorem![]() |
Classification | msc 12-00 |