Discrrete mathematics for computer science digraphs and relations warmup

Digraphs and Relations
Warm Up


The Divisibility Relation

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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

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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 aIf b≠0 then aIf b=0 then NOT b|a since 0|a only if a=0.


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So “|” is a partial order.
It is not a total order because, for example, neither 2|3 nor 3|2 is
true.


FINIS