Question

a1,a2,a3....a30-each of these 30 sets has 5 elements.b1,b2,....bn-each of these n sets has 3 elements.union of a1,a2...a30=union of b1,b2....bn=S.
if each elements of S is in 10 a sets and 9 bsets.then n=?
ans=45.
show the process

Answer 1

ok, here we go~

So the number of elements in the union of the A sets is:
\[5(30)-r_A\]where r is the number of repeats.
Likewise the number of elements in the B sets is:
\[3n-r_B\]
Each element in the union (in S) is repeated 10 times in A, which means if x was the real number of elements in A (not counting repeats) then 9 out of those 10 should be thrown away, or 9x.
Likewise on the B side, 8x of those elements should be thrown away. so now we have:
\[150-9x = 3n-8x \iff 150-x = 3n \iff 50-\frac{x}{3} = n\]
Now, to figure out what x is, we need to use the fact that the union of a group of sets contains every member of each set. if every element in S is repeated 10 times, that means every element in the union of the A's is repeated 10 times. This means that:
\[\frac{150}{10} = 15\]is the number of elements in the the A's without repeats counted (same for the Bs as well).
So now we have:
\[50-\frac{15}{3} = n \iff n = 45\]

Answer 2

i understand.r u a teacher?

Answer 3

im an undergrad.