Arthur D. answered • 02/09/15

Effective Mathematics Tutor

draw 2 intersecting circles inside a rectangle

the circle on the left is A

the circle on the right is B

the rectangle is the universe U

U contains 32 elements

A contains 11 elements

B contains 17 elements

AUB contains 25 elements

this means there are 7 elements (32-25) not in A or B (32 elements in the universe)

the number of elements in A is 11 and the number of elements in B is 17

if A and B were disjoint, not intersecting, then AUB would contain 28 elements

but because AUB contains 25 elements, that means A and B intersect and there are 28-25=3 elements in A∩B

there are 8 elements in A only, 3 elements in A and B (their intersection), and 14 elements in B only

(8+3+14=25 which is how many elements in AUB)

n(AUB') means all the elements in the universe that are not in B

there are 8 in A only and 7 in the universe that are not in A or B for a total of 15 elements in AUB'

(you can't use the 3 in the intersection because they are in B as well as A and you can't use the 14 that are in B)

n(A'∩B')=n(AUB)' which is one of DeMorgan's Laws

n(AUB)=25 so n(AUB)'=everything in the universe not in A or B which is 7 elements (remember that n(U)=32 and AUB=25 so 32-25=7

n(A∩B)=3 which we already determined from above