Question

Consider the poset  $(\mathbb{Z}+, \leq )$ where for a and b in $(\mathbb{Z}+ )$ ,   a <  b if and only if a|b.

The join of a and b is their least common multiple and the meet of a and b is their greatest common divisor:

a ⋁ b = LCM(a,b) and a ⋀ b =GCD(a,b)

Verify the associative property holds for Part B:

As an example, answer for A are shown below:

Answer 1

Solution of claim:- ancbnc) = canbinc definition of GIB, By we have and ancbnc) < a anCbC) ? AC Moreover bne <b . & bncec By transitive property - ancone) a baczb & anebno) abnesc of ancbnd2C E ancbnc) < b This anebne) is up lower bound of a bloc so by definition of greatest epper bound we have O ancone) < and since ancone) is lower bound bof anb fc = anebne) a cand) ne similarly canb) nes and f (amb) accc moreover andra a anbab 1. By transitive property

Саль) лс <aль а4 (аль улс <aль <ь Thys (anbonc is lower bound of bfc so by definition of greatest lower bound . we have (аль )лс - bлс since carbone is gor lower bound of at bnc — 0 = (аль) па кальлі) fooo o 4 Ф Галсьлс ) = (аль) лс / Hence pored