Homewrok set # 1 solution posted on 9/11/02
Problem 1) Take }5,4,3,2,1,0{=A
a) }5,,,/),,{(
=
+
+
∈∈∈= zyxAzAyAxzyxS
2) [GAR] 2.3
a) }12,11,10,9,8,7,6,5,4,3,2{=S
b) }12,10,8,6,4,2{=A
c) if )}1,1{(2 =⇒= Ssum
)}1,2(),2,1{(3 =⇒= Ssum
In general for 1
min(6 , 1)
(1, 6)
{( , )}
k
Max k
sum k S j k j
−
−
=⇒ = −
∪12
≤
≤
k
For example if )}2,6(),3,5(),4,4(),5,3(),6,2{(8 =⇒= Ssum
*Lower bound starts from 2 *Upper bound goes to 6
3) [GAR] 2.7
a) )}1,2,3(),1,3,2(),2,3,1(),2,1,3(),3,1,2(),3,2,1{(=S
b) )}2,3,1(),3,2,1{(
1=A)}1,2,3(),3,2,1{(
2=A)}3,1,2(),3,2,1{(
3
=
A
c)
)}3,2,1{(
321 =AAA ∩∩
The number of every ball corresponds to the number of the draw.
d)
)}3,1,2(),1,2,3(),2,3,1(),3,2,1{(
321 =AAA ∪∪
The number of at least one ball corresponds to the number of the draw.
e) (
)}1,3,2(),2,1,3{()
321 =
′
AAA ∪∪
None of the balls has a number corresponding to the number of the draw.
4) [GAR] 2.12
)()( BABA ′′ ∩∪∩
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