Chapter 8 Study Set: Relations

Transitive Closure

- Add ordered pairs that make a non-transitive relation transitive. Adding the least number of ordered pairs to ensure transitivity is called the *transitive closure* of the relation. - See slides 45-48 for example

Directed Graph

A relation can be on the same set, which we must use a directed graph to draw.

Symmetry

For each pair of elements, each element points to the other. - if (x,y), then (y,x)

Transitivity

For each triple of elements, there is a "triangle" - for a,b,c if we have (a,b) and (b,c), then we have (a,c)

Reflexivity

If each element points to (pairs with) itself in R then R is a reflexive relation. - Slide 24

Properties of infinite sets

See SLides 32-40 for proofs on infinite sets********

Define Relations on Sets

To define a relation Q from R to R - For all real numbers x and y, - x Q y <=> x = y

Inverse of a Relation

if R is a relation from set A to set B, then a relation R-1 from B to A is the inverse relation of R - R-1 = { (y,x) in B x A | (x,y) in R}