By Franklin Mendivil

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**Extra resources for Fractals, Graphs, and Fields**

**Example text**

2 does not have a multiplicative inverse). Properties (i), (iii), and (iv) are the defining properties for an algebraic system with one binary operation, denoted here by +, called a group. If property (ii) also holds then we have a commutative group. By properties (v)–(viii) the nonzero elements of a field form a commutative group under the binary operation of multiplication. In the next theorem we collect a number of elementary properties of fields whose proofs are straightforward. 2 Let F be a field.

Xn ). Thus the number of such r-combinations equals (n + r − 1)! n+r−1 n+r−1 = = . (n − 1)! r n−1 Another useful counting technique is provided by the inclusionexclusion formula. Let X1 , X2 , . . , Xn be subsets of a finite set U. Then the number of elements of U in none of the sets X1 , X2 , . . ,n};|K|=k | ∩i∈K Xi |. Here Xi is the complement of Xi in U, that is, the subset of elements of U that are not in Xi . For the value k = 0 in the formula, we have K = ∅, and ∩i∈∅ Xi is an intersection over an empty set and is interpreted as U.

2 does not have a multiplicative inverse). Properties (i), (iii), and (iv) are the defining properties for an algebraic system with one binary operation, denoted here by +, called a group. If property (ii) also holds then we have a commutative group. By properties (v)–(viii) the nonzero elements of a field form a commutative group under the binary operation of multiplication. In the next theorem we collect a number of elementary properties of fields whose proofs are straightforward. 2 Let F be a field.

### Fractals, Graphs, and Fields by Franklin Mendivil

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