discrete_mathematics_with_applications/chapter_8/notes.md

Page 512

Definition

Let R be a relation from A to B. Define the inverse relation R^{-1} from B to A as follows:

 R^{-1} = \{(y, x) \in B \times A | (x, y) \in R\} 

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Page 513

Definition

A relation on a set A is a relation from A to A.

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Page 514

Definition

Given sets A_1, A_2, \dots, A_n an $n$-ary relation R on A_1 \times A_2 \times \cdots \times A_n is a subset of A_1 \times A_2 \times \cdots \times A_n. The special cases of $2$-ary, $3$-ary, and $4$-ary relations are called binary, ternary, and quarternary relations, respectively.

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Page 518

Definition

Let R be a relation on a set A.

1. R is reflexive if, and only if, for every x \in A, x R x.

2. R is symmetric if, and only if, for every x, y \in A, \text{ if } x R y \text{ then } y R x.

3. R is transitive if, and only if, for every x, y, z \in A, \text{ if } x R y \text{ and } y R z \text{ then } x R z.

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Page 523

Proof of Reflexivity:

Suppose m is a particular but arbitrarily chosen integer. [We must show that m T m.] Now m - m = 0. But 3 | 0 since 0 = 3 \cdot 0. Hence 3 | (m - m). Thus, by definition of T, m T m [as was to be shown].

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Page 524

Proof of Symmetry:

Suppose m and n are particular but arbitrarily chosen integers that satisfy the condition m T n. [We must show that n T m.] By definition of T, since m T n then 3 | (m - n). By definition of "divides", this means that m - n = 3k, for some integer k. Multiplying both sides by -1 gives n - m = 3(-k). Since -k is an integer, this equation shows that 3 | (n - m). Hence, by definition of T, n T m [as was to be shown].

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Page 524

Proof of Transitivity:

Suppose m, n, and p are particular but arbitrarily chosen integers that satisfy the condition m T n and n T p. [We must show that m T p.] By definition of T, since m T n and n T p, then 3 | (m - n) and 3 | (n - p). By definition of "divides", this means that m - n = 3r and n - p = 3s, for some integers r and s. Adding the two equations gives (m - n) + (n - p) = 3r + 3s, and simplifying gives that m - p = 3(r + s). Since r + s is an integer, this equation shows that 3 | (m - p). Hence, by definition of T, m T p [as was to be shown].

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Page 525

Definition

Let A be a set and R a relation on A. The transitive closure of R is the relation R^t on A that satisfies the following three properties:

1. R^t is transitive.

2. R \subseteq R^t.

3. If S is any other transitive relation that contains R, then R^t \subseteq S.