Maths(二)---Matrix Arithmetic and Relations(３)

(a) Reflexive Property of a Binary Relation

Definition:
A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.

例一.
Consider the following relations on {1, 2, 3, 4}:
R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
R6 = {(3, 4)}.
Which of these relations are reflexive?

The relations R3 and R5 are reflexive since they both contain all pairs of the form (a,a), namely, (1, 1), (2, 2), (3, 3) and (4, 4).

(b) Symmetric Property of a Binary Relation

Definition:
A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R,for a, b ∈ A.

A relation R on a set A such that (a, b) ∈ R and (b, a) ∈ R only if a = b, for a, b∈ A, is called anti-symmetric.


例一.

Which of the relations below are symmetric and which are anti-symmetric?R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
R6 = {(3, 4)}.

The relation R2 and R3 are symmetric
R4, R5, R6 are all anti-symmetric. For each of these relations there is no pair ofelements a and b with a ≠ b such that both (a, b) and (b, a) belong to the relation

(c) Transitive Property of a Binary Relation

Definition:
A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, for a, b, c ∈ A.

例一.
Which of the relations bellow are transitive?
R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
R2 = {(1, 1), (1, 2), (2, 1)}
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}


R4 and R5 are transitive.
For instance,

R4 is transitive, since (3, 2) and (2, 1), (4, 2) and (2, 1), (4, 3) and(3, 1), and (4, 3) and (3, 2) are the only such sets of pairs, and (3, 1), (4, 1), and(4, 2) belong to R4