Basic Properties of Binary Relations on a Set Reflexivity, Symmetry, Transitivity Basic properties: Let R be a binary relation on a set A. R is reflexive, iff for all x A, (x,x) R, i.e. xRx is true.   R is symmetric, iff for all x, y A, if (x, y) R, then (y, x) R    i.e xRy yRx is true  R is transitive iff for all x, y, z A, if (x, y) R and (y,z) R , then (x, z) R     i.e. (xRy yRz) xRz is true   Reflexive relations Definition Let R be a binary relation on a set A. R is reflexive, iff for all x A, (x,x) R, i.e. xRx is true.   Examples: Equality is a reflexive relation
Basic Properties of Binary Relations on a Set Reflexivity, Symmetry, Transitivity Basic properties: Let R be a binary relation on a set A. R is reflexive, iff for all x  A, (x,x)  R, i.e. xRx is true. R is symmetric, iff for all x, y  A, if (x, y)  R, then (y, x)  R i.e xRy  yRx is true R is transitive iff for all x, y, z  A, if (x, y)  R and (y,z)  R , then (x, z)  R i.e. (xRy  yRz)  xRz is true Reflexive relations Definition Let R be a binary relation on a set A. R is reflexive, iff for all x  A, (x,x)  R, i.e. xRx is true. Examples: Equality is a reflexive relation
MFCS.docx