Cartesian product of sets:
The Cartesian product (also called as the cross product) of two sets A and B, shown by A x B (in the similar order) is the set of all ordered pairs (x, y) such that x∈A and y∈B. What we denote by ordered pair is that the pair(a, b) is not the similar pair as (b, a) unless a = b. It implies that A x B ≠ B x A in common. Also if A contains m kind of elements and B contains n elements then A x B have m x n elements.
Same as we may define A x A = {(x, y); x∈A and y∈A}. We may also describe cartesian product of more than two sets.
e.g. A1x A2 x A3 x . . . .´ An = {(a1, a2, . . . , an): a1 ∈A1, a2 ∈ A2, . . . , an ∈ An}
Problem: If A ={a, b, c} and B = {b, c, d} then calculate
(i). A u B , A ∩ B , A-B and B-A
(ii). A x B and B x A
Solution: (i) A u B = {x: x∈A or x∈B}= {a, b, c, d}
A ∩ B = {x: x∈A and x∈B}= {b, c}
A-B = {x: x∈A and x∉B}= {a}
B-A = {x: x∈B and x∉B}= {d}
(ii) A x B = {(x, y): x∈A and y∈B}
= {(a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, b), (c, c), (c, d)}
B x A = {(x, y): x∈B and y∈A}
= {(b, a), (b, b), (b, c),(c, a), (c, b), (c, c),(d, a), (d, b), (d, c)}
Note that A x B ≠ B x A.
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