If set
S
is a non-empty subset of integer set
Z
, if
∀
a
,
b
∈
S
,
a
b
∈
S
, then
S
is
closed under multiplication
.
Given that for set
T
,
U
:
T
⊆
Z
,
V
⊆
Z
,
T
∩
V
=
∅
,
T
∪
V
=
Z
, and
∀
a
,
b
,
c
∈
T
,
a
b
c
∈
T
,
∀
x
,
y
,
z
∈
V
,
x
y
z
∈
V
, then which of the choices is true?
A
.
At least one of T,V is closed under multiplication.
B
.
At most one of T,V is closed under multiplication.
C
.
Only one of T,V is closed under multiplication.
D
.
Both T,V are closed under multiplication.
Have a look at my problem set:
SAT 1000 problems
Consider two examples that fit the given requirements.
Example 1: T is all the odd numbers, V is all the even numbers. Both are closed under multiplication.
Example 2: T is all the odd numbers except ± 3 , V is all the even numbers and ± 3 . ( V still meets the requirement ∀ x , y , z ∈ V , x y z ∈ V because even if − 3 and 3 are chosen as two of the numbers, the third number will make the product even.) T is closed under multiplication (without 3 , no two odds can multiply to 3 ), but V is not closed under multiplication because − 3 ⋅ 3 = − 9 , and − 9 ∈ / T .
In the first example both are closed, but the second example only one is closed, so out of the given choices we can conclude that A) at least one of T , V is closed under multiplication .