Q: Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find:
a.) A x B x C
b.) C x B x A
c.) C x A x B
d.) B x B x B
a.) A x B x C = [{a, x, 0}, {a, y, 1}, {b, x, 0}, {b, y, 1}, {c, x, 0}, {c, y, 1}]
b.) C x B x A = [{0, x, a}, {0, x, b}, {0, x, c}, {1, y, a}, {1, y, b}, {1, y, c}]
c.) C x A x B = [{0, a, x}, {0, b, x}, {0, c, x}, {1, a, y}, {1, b, y}, {1, c, y}]
d.) B x B x B = [{x, x, x}, {x, x, y}, {x, y, x}, {y, y, y}, {y, y, x}, {y, x, y}]
__________
How about now Mush ( is everything correct now):
A x B x C = [{a, x, 0}, {a, x, 1}, {a, y, 0}, {a, y, 1}, {b, x, 0}, {b, x, 1},
{b, y, 0}, {b, y, 1}, {c, x, 0}, {c, x, 1}, {c, y, 0}, {c, y, 1}]
C x B x A = [{0, x, a}, {0, x, b}, {0, x, c}, {1, y, a}, {1, y, b}, {1, y, c}]
C x A x B = [{0, a, x}, {0, b, x}, {0, c, x}, {1, a, y}, {1, b, y}, {1, c, y}]
B x B x B = [{x, x, x}, {x, x, y}, {x, y, x}, {y, y, y}, {y, y, x}, {y, x, y}]
a) is now complete.
b) is not complete, for example, (1,x,c) is not there.
c) is not complete, for example, (0,a,y) is not there.
d) is not complete, for example, (x,y,y) is not there.
Hint: a), b) and c) should have the same number of elements. d) should have 8 elements.
PS your notation is wrong. Your sets A x B x C should be sets of ordered-3tuples, not sets of sets! Curly brackets denote sets, normal brackets denote ordered n-tuples:
a) should be:
Mush, hopefully 3rd times a charm:
A x B x C = {(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0),
(b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)}
C x B x A = {(0, x, a), (0, x, b), (0, x, c), (1, x, a), (1, x, b), (1, x, c), (0, y, a),
(0, y, b), (0, y, c), (1, y, a), (1, y, b), (1, y, c)}
C x A x B = {(0, a, x), (0, b, x), (0, c, x), (1, a, x), (1, b, x), (1, c, x), (0, a, y),
(0, b, y), (0, c, y), (1, a, y), (1, b, y), (1, c, y)}
B x B x B = {(x, x, x), (x, x, y), (x, y, x), (x, y, y), (y, y, y), (y, y, x), (y, x, y),
(y, y, x)}