set theory

Question

STATE TRUE OF FALSE

The Poset [ D_n,/ ] is a distributive lattice for any positive integer n, where D_n stands for divisors of n and relation "/" is a divides operation (i.e) (a,b) belongs to relation,R iff a divides b.

Please support your answer by giving explanations.

Answer 1

solution

Answer 2

Yes, True. The set $D_n$ of all positive integer divisors of a fixed integer $n$, ordered by divisibility, is a distributive lattice.

If You seek a Formal Proof, Here it goes :

We already know that "The set $(D_n,/)$ is a  lattice." (also Can be proved easily)

So, In Order to Prove it Distributive, We need to Prove either one of the Two Distributive rules (Because If One holds then Other also Holds)

So, We need to Prove that $a \vee (b \wedge  c) = (a \vee b) \wedge (a \vee c)$