Thread: sets and relations #2

Let A={1,2,4,6,8} and for a ,b, Element of A, define a < b if and only if b/a is an integer.
(a) prove that < defines a partial order on A
(b) draw the Hasse diagram for <
(c) List all minimum, minimal, maximum, and maximal elements.
(d) Is (A, <) totally ordered? explain

Originally Posted by tygracen

Let A={1,2,4,6,8} and for a ,b, Element of A, define a < b if and only if b/a is an integer.
(a) prove that < defines a partial order on A

what does the definition of "partial order" say?

you have to show that reflexivity, antisymmetry and transitivity holds

see: Partial Order -- from Wolfram MathWorld

(b) draw the Hasse diagram for <

see here

(c) List all minimum, minimal, maximum, and maximal elements.

again, what you need to do here becomes clear when you write out the definitions, which i won't do since that's reinventing the wheel. check your text, or google, or wikipedia, or ....

(d) Is (A, <) totally ordered? explain

in addition to being a partial order, can any two elements be compared?

see: Totally Ordered Set -- from Wolfram MathWorld