how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?
thanks in advance
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how do i prove this lattice (N0,Lowest Common Multiple (LCM) ,great common divisor (GCD)) is distributive?
thanks in advance
What does that mean?Quote:
Originally Posted by JK64
be L a lattice
L=(N0,LCM,GCD)
in which N0= is da naturals set with zero included
LCM is the Lowest Common Multiple
GCD is the great common divisor
And i want need to prove that lattice L is dristibutive which is
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).
In this case i need to prove,
GCD(a,LCM(b,c)) == LCM(GCD(a,b),GCD(a,c))
I've already starting thinking about 4 possible cases,
i) a|/b & a|/c [which is easy to prove]
ii) a|b & a|/c
iii) a|/b & a|c
iv) a|b & a|c
ii) & iii) i think i'm close to the answer, but i'm really struggling with iv)
about the iv)
i'm thinking about this:
If a|b & a|c then
(i) GCD(a,LCM(b,c))=a
on other hand
(ii) LCM( GCD(a,b) , GCD(a,c) )=LCM(a,a)=a
hence, (i)==(ii), what do you think?