Let the universal set S have 176 elements.
Sets A and B are subsets of S.
Set A contains 93 elements, and Set B contains 35 elements.
If the total number of elements
in either Set A or Set B is 127,
how many elements are in Set B but not in Set A?
To answer the question,
you'll first complete the Venn diagram shown below.
Because universal set has 176 elements,
we know the cardinality of Set S is equal to 176.
Because Set A contains 93 elements,
we know the cardinality of Set A is 93,
and we know the cardinality of Set B is 35.
Next we're also told, the number of elements
in either Set A or Set B is 127.
The set of elements in Set A or Set B is A union B,
and therefore the cardinality of A union B
is equal to 127.
Now to help us complete the Venn diagram,
the next step will be to determine the number of elements
in the intersection of A and B
using the formula shown below.
Where the cardinality of A intersect B
is equal to the cardinality of A plus the cardinality of B
minus the cardinality of A union B.
So using the formula,
we have the cardinality of A intersect B
is equal to the cardinality of A which is 93
plus the cardinality of B, which is 35,
minus the cardinality of A union B, which is 127.
Simplifying, 93 plus 35 is 128.
Then we have minus 127, which is equal to one.
So now we know the number of elements in A intersect B,
this set here,
is equal to one.
Now let's complete the Venn diagram.
We know the number of elements in Set A is equal to 93,
so there must be 93 elements in this set here.
We already have one element here, 93 minus one is 92,
and therefore, there are 92 elements that are only in A
and one element that's in Set A but also in Set B.
We also know the number of elements in Set B is 35.
Well, here's Set B.
Again, there's already one element in the intersection,
and therefore, the number of elements that are only in B
is going to be 35 minus one, which is 34.
It doesn't ask,
but let's also determine the number of elements
that are in the universal set but not in Set A or Set B.
We know there are 176 elements in the universal set.
So if we take 176
and subtract the sum of these cardinalities,
we can determine the number of elements in the universal set
but not in Set A or Set B.
176 minus this sum is equal to 49,
and therefore, there are 49 elements
in the universal set that are not in Set A or Set B.
Now let's finally answer the question.
The question is, how many elements are in Set B
but not in Set A?
So here is Set B.
We're looking for the elements that are in Set B
but not in Set A, which is this set here.
This is the set of elements that are in Set B
but not in Set A, and therefore the answer is,
there are 34 elements in Set B but not in Set A.
I hope you found this helpful.