Suppose Set B contains 69 elements and the total number elements in either Set A or Set B is 107. If the Sets A and B have 13 elements in common, how many elements are contained in set A?
Union is 107
Intersection is 13
draw Venn diagram
69 + A alone = 107
so
A alone = 38
plus A and B = 38+13 = 51
Well, if we know that Set B contains 69 elements and the total number of elements in either Set A or Set B is 107, then we can start by subtracting the number of elements in Set B from the total.
So, 107 - 69 = 38.
Now, since Sets A and B have 13 elements in common, we can subtract that from the 38 to find the remaining elements in Set A.
38 - 13 = 25.
Therefore, there are 25 elements in Set A.
To determine the number of elements in set A, we need to use the principle of inclusion-exclusion.
Let's denote:
n(A) as the number of elements in set A.
n(B) as the number of elements in set B.
n(A ∪ B) as the number of elements in either set A or set B.
n(A ∩ B) as the number of elements in both set A and set B.
According to the principle of inclusion-exclusion, the formula is:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
Given the following information:
n(B) = 69
n(A ∪ B) = 107
n(A ∩ B) = 13.
We can plug in the values and solve for n(A):
107 = n(A) + 69 - 13.
Rearranging the equation:
107 - 69 + 13 = n(A).
Performing the calculations:
51 = n(A).
Therefore, set A contains 51 elements.
To find the number of elements in Set A, we need to subtract the number of elements in Set B along with the number of elements common to both sets from the total number of elements in either Set A or Set B.
Given:
Number of elements in Set B = 69
Total number of elements in either Set A or Set B = 107
Number of elements common to both Set A and Set B = 13
To find the number of elements in Set A, we can use the formula:
Number of elements in Set A = Total number of elements in either Set A or Set B - Number of elements in Set B - Number of elements common to both sets
Substituting the given values:
Number of elements in Set A = 107 - 69 - 13
Calculating:
Number of elements in Set A = 35
Therefore, Set A contains 35 elements.