Is -4 in S? from DS 1000 series

[mehravikas]

S is a set of integers such that,

1. If a is in s, then -a is in S.
2. If each of a and b is in S, then ab is in S.

Given that -

1. 1 is in S
2. 2 is in S

Is -4 in S?

[jasonc]

answer should be b.

given:
S:{a, -a, -a^2, +/-a^3, +/-a^4.....}
(if you dont' understand the above, read (-a) as b and it should make sense)

stem1: 1 is in S => S has at least 1 & -1
INSUFF

stem2: 2 is in S => S has at least 2, -2, -4, +/-2^n where n is any positive integer
SUFF

[mehravikas]

answer should be b.

given:
S:{a, -a, -a^2, +/-a^3, +/-a^4.....}
(if you dont' understand the above, read (-a) as b and it should make sense)

stem1: 1 is in S => S has at least 1 & -1
INSUFF

stem2: 2 is in S => S has at least 2, -2, -4, +/-2^n where n is any positive integer
SUFF

Your answer is correct, but I am still not able to understand the way you have made a series of values in the set. From what I understood, the set should be something like S = {a, -a, b, ab}

Can you explain further, please?

[jasonc]

I think the point you may be missing is that the original 2 statements apply to ALL elements of S. Meaning that if 'a' and '-a' are in the set, then 'ab', or in this case 'a' & '-a', must also be in the set.

like I hinted at in my original post, substitue -a in for b.

[mehravikas]

I think the point you may be missing is that the original 2 statements apply to ALL elements of S. Meaning that if 'a' and '-a' are in the set, then 'ab', or in this case 'a' & '-a', must also be in the set.

like I hinted at in my original post, substitue -a in for b.

Getting it now )

Thanks,

Vikas

[airan]

JasonC,
How can we assume -a=b, b can be something else also.
For example:
if a=2, b=1 that means, -2 is also in set but how b=-2 ?

I think it is based on assumption that b=-a, otherwise the answer should be E. Can u pls try to explain it once more(why assume b=-a)

[zacharyz]

From (2), you know that S contains at least {2}. There is no other guarantee that anything else is in the set... but let's look at the rules:

i) says that if a is in the set, then -a is in the set.
Now you know that -2 is also in the set.

ii) says that if a and b are in the set, then a*b are in the set.
We know that S at least contains {2, -2} now we have an a and a b. In this case, a = 2 and b = -2. Now you know that a*b is in the set, by definition, and therefore -4 is in the set.

So (B) is sufficient.

Other observations, S is going to become an infinite set in 2 as S becomes {2, -2, -4, 4, 8, -8, 16, etc...}

Statement 1 only guaranteed that S is {1, -1}... and then unknowns

Just remember that a and b are not specific numbers, but any of two different variables.