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

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What is the number of integers that are common to both set S and set T [#permalink]
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eybrj2
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

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

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Re: What is the number of integers that are common to both set S and set T [#permalink]
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eybrj2
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

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
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Re: What is the number of integers that are common to both set S and set T [#permalink]
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eybrj2
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

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

in B it states: "in both" - aren't these the common integers? can you please help me understand?
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Re: What is the number of integers that are common to both set S and set T [#permalink]
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eybrj2
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
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
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Re: What is the number of integers that are common to both set S and set T [#permalink]
omavsp
TestPrepUnlimited
eybrj2
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

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

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

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?
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Re: What is the number of integers that are common to both set S and set T [#permalink]
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.

Regards.

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Re: What is the number of integers that are common to both set S and set T [#permalink]
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Bunuel
What is the number of integers that are common to both set S and set T?

Look at the diagram:
Attachment:
Sets.PNG
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.

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."?
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Re: What is the number of integers that are common to both set S and set T [#permalink]
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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?
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What is the number of integers that are common to both set S and set T [#permalink]
achloes
If we consider the case that U is the set of integers that are in S only
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.

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Re: What is the number of integers that are common to both set S and set T [#permalink]
achloes
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?

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.
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Re: What is the number of integers that are common to both set S and set T [#permalink]
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?

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Re: What is the number of integers that are common to both set S and set T [#permalink]
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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

(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.
Re: What is the number of integers that are common to both set S and set T [#permalink]
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