Question

Let U be set with number of elements in it is 2009 and A, B are subsets of U with n (A U B) = 280. If
n(A' B') = x1^3+ x2^3 = y1^3 + y2^3
for some positive integers x1, < y1, < y2 < X2, then find value of
X₂ + y2 / x1 + y1
​

Answer 1

Answer:According to the problem,

n(U)=2009.....(1) and n(A∪B)=280.......(2).

Now

n(A

′

∩B

′

)

=n{(A

′

∩B

′

)}

=n({(A∪B)

′

} [Using De Morgan's law]

=n(U)−n(A∪B)

=2009−280

=1729.

Now 1729 can be expressed as 10

3

+9

3

and 12

3

+1

3

.(For this particular property of 1729, it is called Ramanujan's Number.)

Comparing with the given problem we have

x

1

​

=10,x

2

​

=9 and y

1

​

=12,y

2

​

=1.

∴(x

1

​

+x

2

​

+y

1

​

+y

2

​

)=(10+9+12+1)=32.

Step-by-step explanation: