Question

Va rog mult !! Exercitiul din poza

Answer 1

Explicație pas cu pas:

{x}=x-[x]

a.{2x}=2x-[2x]=0

[2x]=2x ⇒x∈Z

b. {x}=1/3

x-[x]=1/3

[x]=x-1/3

[x]=(3x-1)/3=k , [x]≤x<[x]+1 , x=(3k+1)/3

k≤(3k+1)/3<k+1

3k+1≥3k⇒k∈Z

3k+1<3k+3⇒k∈Z

(3k+1)/3∈Z

3k+1∈M(3)⇒3k+1∈{0,3,6,9,,,3n} ,n∈N

⇒3k∈{-1,2,5,8,,,3n-1}⇒k∉Z⇒x∉R⇒x∈∅