S, have 176 elements. A and B are subsets of S. Set A contains 71 elements and Set B contains 80 elements. If Sets A and B have 11 elements in common, how many elements are in neither A nor B?
You have a total of 71+80-11 = 140 in A and B
So, 36 are in neither.
To find the number of elements that are neither in set A nor in set B, we can use the principle of inclusion-exclusion.
First, let's find the number of elements that are either in set A or in set B. We add the number of elements in set A (71) with the number of elements in set B (80):
71 + 80 = 151
However, by doing this, we have counted the 11 elements that are in both set A and set B twice. To correct for this, we need to subtract the number of elements that are in both sets once.
So we subtract the number of elements in the intersection of set A and set B (11) from the total number of elements in the union of set A and set B (151):
151 - 11 = 140
Therefore, there are 140 elements that are either in set A or in set B.
Now, since set S has a total of 176 elements, we can find the number of elements that are neither in set A nor in set B by subtracting the number of elements in the union of set A and set B (140) from the total number of elements in set S (176):
176 - 140 = 36
Therefore, there are 36 elements that are in neither set A nor set B.