What is the number of integers that are common to both set S and set T

Question

What is the number of integers that are common to both set S and set T ?

(1) The number of integers in S is 7, and the number of integers in T is 6.
(2) U is the set of integers that are in S only or in T only or in both, and the number of integers in U is 10.

DS41402.01

Bunuel · Accepted Answer

What is the number of integers that are common to both set S and set T?

Look at the diagram:

Attachment:

Sets.PNG [ 2.79 KiB | Viewed 21674 times ]

We are asked to find the intersection of the sets, the # of integers that are both in S and T (yellow area).

(1) The number of integers in S is 7, and the number of integers in T is 6. There can be from 0 to 6 common integers in sets S and T. Not sufficient,

(2) U is the set of integers that are in S only or in T only or in both, and the number of integers in U is 10. U=10 is the sum of blue, yellow or red areas. Not sufficient.

(1)+(2) Now, Total={S}+{T}-{Both} --> 10=7+6-{Both} --> {Both}=3. Sufficient.

Answer: C.

boomtangboy · Answer

Hi, +1 C The easier way to attack this problem is through the use of venn diagrams but i will try to make this as clear as possible without them

Statement 1 : Just says S = 7 & T= 6 no information of how many elements common or arre there in the universal set Hence Insufficient Statement 2 : Set U has all the elements of S & T including the common ones & beyond this there are no elements in the universe. So basically this set is your 2 big circles (Venn Diagram) incl the middle portion Together : You know from Statement 2 that universe only has elements from U => S & T You know the elements of S & T thus the middle overlapping portion = S+T-Common = Universe => 7+6- Common = 10 Common = 3 Hence C is the answer... Hope this helps :-D

reynaldreni · Answer

Clearly, statements 1 or 2 are not sufficient on their own. 
Let S = elements only in Set S
Let X = elements in both S & T 
Let T = elements only in Set T
Combining the statement together, we get
S+X = 7 & 
T+X = 6
S+T+2X = 13 --(1)

U = 10
U = S+T+X
S+T+X = 10 --(2)

(1) - (2), we get 
X = 3

OA <- C

TestPrepUnlimited · Answer

Combined:

If S and T have all unique integers, then the combined set U should have 6 + 7 = 13 integers. However according to (2) we only have 10 integers, this is because some of them must be shared. Since we have 3 less than 13, this means those 3 numbers were shared. Thus there are 3 integers common between S and T.

Ans: C

omavsp · Answer

in B it states: "in both" - aren't these the common integers? can you please help me understand?

Pritishd · Answer

Let,
 x   =  Elements in S only
 y   =  Elements in T only
 z   =  Elements in both S and T (The question asks us to find the value of  z )

Statement - I ( Insufficient ) 
1.  x + z = 7  and  y + z = 6 
2. We cannot solve for  z  since the we have  2  equations and  3  variables

Statement - II ( Insufficient ) 
1.  x + y + z = 10 
2. Again, we cannot solve for  z  since the we have  1  equation and  3  variables

Combined ( Sufficient ) 
1. Add the  2  equations from statement-I ->  x + y + 2z = 13 
2. From the equation from statement-II we have  x + y = 10 - z 
3. Plugging in  2  into  1  ->  10 - z + 2z = 13  ->  z = 3

Ans.  C

Pritishd · Answer

Not sure if I completely understand your doubt but we need a unique value of the number of integers common to both  S  and  T . Even though B mentions ' in both ' it does not give you the value for the same.

In other words  S only + T only + Both = 10 . Do you think we can get a unique value for  Both  from this equation?

Basim2016 · Answer

Respected Bunuel,
 
I got it wrong as I thought that it is nowhere mentioned that Set U has all the elements of S and T. So,consequently, U has 3 elements of S and all remaining 7 from set T. SO, I marked it E.

PLEASE guide.

Regards.

Posted from my mobile device

MHIKER · Answer

As I understand 10 is the sum of the Total. So, will not the formula be  Total = S+T-Both+Neither.  Can we determine both if we don't have the "Neither."?

achloes · Answer

avigutman   chetan2u   Bunuel

I'm confused about the use of "or" in statement 2 because it creates 3 possible scenarios with different outcomes each.

For example, if we consider the case that U is the set of integers that are in S only, how can we conclude anything about the remaining 3 integers in set U? What if those fall in the "neither" bucket?

avigutman · Answer

But that’s not a case,  achloes . If I say that I want to travel to either Europe or Africa, it would be wrong to say that I only want to travel to Europe. 
The statement tells us that if we count all the integers in the three groups (T but not S, S but not T, and both S and T) we’ll see 10 of them.

Posted from my mobile device

chetan2u · Answer

Hi

It means that if you pick up an integer from that set, it will fall in any of the three categories and vice versa, meaning the number of integers in the set will be equal to the number of integers in all three categories added together.

TargetMBA007 · Answer

I did this using the formula

(AorB) = A + B - (A&B).
(A&B) = A + B - (AorB).
(A&B) = 7+6-10 = 3

and it got me the right answer

But, I found the wording a bit confusing, so I wasn't sure, if this is the right formula to use here.
When it says "U is the set of integers that are in S only or in T only or in both, and the number of integers in U is 10."
For me the A or B part, should only "S Only or T only" - is the (AorB) element of the formula - "or in both - This is the bit that confused me, as this is the part that we are solving for, so I wonder how this can be part of the "AorB" element?

KarishmaB

KarishmaB · Answer

(A or B) is the same as (A union B) which is (A U B) - basically it includes all elements that are in A only, all elements that are in B only and all elements that are in both A and B. Basically, when we look at the Venn diagram discussed in the video below, it is the blue, yellow and green region together. https://youtu.be/HRnuURqGhmg Hence, this formula works: (AorB) = A + B - (A&B) If you are unsure why we are subtracting (A&B), it is because it is double counted in A + B so we need to subtract it out once so that it is counted once only. Check out the YT video above.

bumpbot · Answer

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What is the number of integers that are common to both set S and set T

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