Let A = {a, b, c}, & B= {1, 2}
a) Determine A x B
b) Determine B x A
c) Does A x B = B x A?
d) Does |A x B| = |B x A| ?
a) To determine A x B, you need to find the Cartesian product of A and B. To do this, pair each element from A with every element from B.
A x B = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}
b) To determine B x A, you need to find the Cartesian product of B and A. Pair each element from B with every element from A.
B x A = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
c) A x B and B x A are not the same. In A x B, the order of elements is (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2). In B x A, the order of elements is (1, a), (1, b), (1, c), (2, a), (2, b), (2, c). The order of the elements changes, so A x B is not equal to B x A.
d) The cardinality of a set refers to the number of elements in that set. To determine |A x B| (the cardinality of A x B), you count the number of elements in A x B. Similarly, to determine |B x A|, you count the number of elements in B x A.
|A x B| = 6 (Counting the 6 tuples in A x B)
|B x A| = 6 (Counting the 6 tuples in B x A)
In this case, |A x B| is equal to |B x A|.