Question

Determinati A inclus in B, daca:
a) A={x∈Z l l1-xl ≤5} si B={x∈Zl lxl >4}
b) A={y∈Z l ly+1l ≥ 2} si B={ t∈Z l l2t+3l <7}

Answer 1

a)
I1-xI≤5
radacina este: 1-x=0⇒  -x=-1⇒  x=1
Cazul I
1-x≤5
-x≤5-1
-x≤4    inmultim -1
x≥4
x∈[4, +α)=S1
cazul II
1-x≤-5
-x≤-5-1
-x≤-6   inmultim -1
x≥6
x∈[-6, +α)=S2
S=S1US2
x∈[-6,4]=S
A={-6,-5,-4,-3,-2,-1,0,1,2,3,4}

IxI>4
cazul I
x>4
x∈(4,+α)=S1
cazul II
x>-4
x∈(-4,+α)=S2
x∈(-4,4)
B={-3,-2,-1,0,1,2,3}

ex.2
Iy+1I≥2
cazul I
y+1≥2
y≥2-1
y≥1
y∈[1,+α]=S1
cazul II
y+1≥-2
y≥-2-1
y≥-3
y∈[-3,+α0=S2
S=S1US2
y∈[-3,1]=S
A={-3,-2,-1,0,1}

I2t+3I<7
cazul I
2t+3<7
2t<7-3
t<4/2
t<2
t∈(-α,2)
cazul II
2t+3<-7
2t<-7-3
t<-10/2
t<-5
t∈(-α,-5)
deci t∈(-5,2)
B={-4,-3,-2,-1,0,1}