Discrete Math Notes:
Chapter 5: Functions and Relations
5.7 Properties of binary relations
Reflexive, R is a binary relation on set A, R is reflexive if and only if for every x E A, xRx.
anti-reflexive, R is anti-reflexive if and only if for every x in the domain of R, it is
not true that xRx.
Symmetric, suppose that R is a relation on set A. R is symmetric if and only if for every pair, x and y E A, xRy if and only if yRx.
anti-symmetric, suppose that R is a relation on set A. R is anti-symmetric if and only if for every pair, x and y E A, if x # y then it cannot be the case that xRy and yRx are both true.
Transitive, suppose that R is a relation on set A. R is transitive if and only if for every three elements, x, y, z E A, if xRy and yRz, then it must also be the case that xRz.
Overview:
The term "anti-reflexive" has no common definition. Every element in the set must not be related to itself for a binary relation to be anti-reflexive. To demonstrate that a relation is not anti-reflexive, all that is required is to prove that there is a specific x A such that xRx is true. R is anti-reflexive if and only if xRx is not true for every x in the domain of R. R is symmetric if and only if xRy if and only if yRx for any pair of x and yE A. R is transitive if and only if, for each of the three items x, y, and z E A, xRy and yRz are likewise tru