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.. • What does that mean

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<b” but that is not right! Why?

If b≠0 then a<b (why?) so it cannot be that b|a.

If b=0 then NOT b|a since 0|a only if a=0.

• So “|” is a partial order

• It is not a total order because, for example, neither 2|3 nor 3|2 is true

FINIS