In each of the following parts, you need to find a family of sets {Ek }_keN such

Question

In each of the following parts, you need to find a family of sets
{Ek }_keN such that E_keR for each keN, that no two sets E_k are equal to each other
and that the given conditions hold.

U_keN E_k=R and INTERSECTION_keN E_k=N

Explanation / Answer

This one is little more tricky than the others. Again we will take E_1 = N E_k = R / {1/k} for k > 1 [here the / is set subtraction]. So E_2 is all the real numbers except 1/2, E_3 is all real numbers except 1/3 etc. Since N is contained in each of the E_k [we never remove any natural numbers since we are removing numbers strictly less than 1 and greater than 0] and E_1 = N the intersection property is satisfied. Its not hard to see the union of the sets is all reals.

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