Discrrete mathematics for computer science digraphs and relations warmup

Digraphs and Relations Warm Up The Divisibility Relation • • • • • • Let “|” be the binary relation on N×N such that a|b (“a divides b”) iff there is an n∈N such that a∙n=b Examples: – 2|4 but not 2|3 and not 4|2 – 1|a for any a since 1∙a=a – What about 0|a? – What about a|0? Show that “|” is a partial order but not a total order What does that mean? Reflexive, transitive, antisymmetric But not true that for any a and b, either a|b or b|a a|b iff for some n∈N, a∙n = b • • • Reflexive? a|a for any a since a∙1=a Transitive? If a|b and b|c, then there exist n, m∈N such that a∙n=b and b∙m=c Then a∙(nm)=c so a|c Antisymmetric? Suppose a|b and a≠b We want to say “then a