Question

9. Let S = {a,b,c,d,e,f,g,h,i,j}.

a. is {{a}, {b, c}, {e, g}, {h, i, j}} a partition of S? Explain.

b. is {{a, b}, {c, d}, {e, f}, {g, h}, {h, i, j}} a partition of S? Explain. c. is {{a, b}, {c, d}, {e, f}, {g, h}, {i, j}} a partition of S? Explain.

Answer 1

A set P is a partition of a set S if and only if :

1) The union of all the elements of P is equal to S.

2) The intersection of any two distinct elements of P always comes out to null.

That is, if we partition set S into three parts that is , then and .

a) This is not a partition of S. Since, d and f are missing, the union of the elements will not be equal to S. [Point 1 is violated]

b) This is not a partition. Since h is being repeated in two distinct elements, the intersection of the elements will not be null. [Point 2 is violated].

c) Yes, this is a partition of S. Since the union of all distinct elements will be equal to S itself and the intersection will be null.