8.2 Reflexivity, Symmetry, and Transitivity

Definition

Let R by a relation a set A.

1. R is reflexive if and only if, for every x ∈ A, x R x.
2. R is symmetric if and only if, for every x, y ∈ A, if x R y then y R x.
3. R is transitive if and only if, for every x, y, z ∈ A, if x R y and y R z then x R z.

Alternatively

1. R is reflexive ↔ for every x in A, (x, x) ∈ R
2. R is symmetric ↔ for every x, y in A, if (x, y) ∈ R then (y, x) ∈ R
3. R is transitive ↔ for every x, y, z in A, if (x, y) ∈ R and (y, z) ∈ R then (x, z) ∈ R

Examples

• x R y ↔ x = y
• x R y ↔ x < y
• m T n ↔ 3 | (m - n)

Definition: Transitive Closure

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 properties:

1. R^t is transitive
2. R ⊆ R^t
3. If S is any other transitive relation that contains R, then R^t ⊆ S

In Class Exercises

• 1. A = {0, 1, 2, 3} and R = {(0, 0), (0, 1), (0, 3), (1, 1), (1, 0), (2, 3), (3, 3)}. Is R reflexive? Is R symmetric? Is R transitive?
• 11. D is the relation defined on R as follows: For every x, y ∈ R, x D y ↔ xy ≥ 0. Is D reflexive? Is D symmetric? Is D transitive? Justify your answers.
• 20. Let X = {a, b, c} and P(X) be the power set of X. A relation E is defined on P(X) as follows: For every A, B ∈ P(X), A E B ↔ the number of elements in A equals the number of elements in B. Is E reflexive? Symmetric? Transitive? Justify your answers.
• 51. A = {0, 1, 2, 3) and R ={(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0)}. Find R^t, the transitive closure of R.