a) A is not equal to B because A has five elements, while B has only three elements.
b) B x C is the Cartesian product of sets B and C, which is the set of all possible ordered pairs (b, c) where b is an element of B and c is an element of C.
B x C = {(d,1), (d,2), (d,3), (e,1), (e,2), (e,3), (f,1), (f,2), (f,3)}
c) The union of set C and the empty set is simply the set C itself.
(C U Ø) = {1, 2, 3}
d) The intersection of set C and the empty set is always the empty set.
(C n Ø) = Ø
e) The number of subsets of a set A is 2^n, where n is the number of elements in set A. In this case, A has five elements, so the number of subsets of A is 2^5 = 32. The subsets of A are:
{}, {a}, {b}, {c}, {d}, {e}, {a,b}, {a,c}, {a,d}, {a,e}, {b,c}, {b,d}, {b,e}, {c,d}, {c,e}, {a,b,c}, {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {b,c,d}, {b,c,e}, {a,b,c,d}, {a,b,c,e}, {a,b,d,e}, {a,c,d,e}, {b,c,d,e}, {a,b,c,d,e}
f) B is not a proper subset of A because both sets have elements d and e in common. A proper subset of A would have all the elements of A, except for d and e.
g) (A n B) is the set of elements that are common to both A and B, which is {d, e}. (A n B) x B is the Cartesian product of this set and B.
(A n B) x B = {(d,d), (d,e), (e,d), (e,e)}
Therefore, the elements of (A n B) x B are (d,d), (d,e), (e,d), and (e,e).