# Cartesian Product of Two Sets ## Cartesian Product of Two Sets

Let A and B be sets. Given a ∈ A and b ∈ B, the object (a, b) is called an ordered pair, with first coordinate a and second coordinate b.

Cartesian product. Suppose we have two sets A and B and we form ordered pairs by taking an element of A as the first member of the pair and an element of B as the second member. The Cartesian product of A and B, written A × B, is the set consisting of all such pairs. The predicate notation defines it as:- A × B =def {<x,y> x ∈ A and y ∈ B}.

• Two ordered pairs are equal if and only if both of their coordinates match. That is, (a, b) = (c, d) if a = c and b = d. 2
• Unlike two element sets, the order of the coordinates in an ordered pair matters: {1, 2} = {2, 1} but (1, 2) 6= (2, 1).

Let A and B be sets. The (Cartesian) product of A and B is the set A × B = {(a, b)| a ∈ A and b ∈ B} of all ordered pairs (a, b) with a ∈ A and b ∈ B.

Cartesian Product of Two Sets

We write A × A = A2. 2 ∅ × B = A × ∅ = ∅. 3 A × B 6= B × A unless A = ∅, B = ∅ or A = B.

If A = {1, 2, 3} and B = {x, y}, then A × B = {(1, x),(2, x),(3, x),(1, y),(2, y),(3, y)}.

Let A,B, C, D be sets.

1. A × B ⊂ C × D iff A ⊂ C and B ⊂ D (if A 6= ∅ and B 6= ∅)
2. A × (B ∩ C) = (A × B) ∩ (A × C)
3. A × (B ∪ C) = (A × B) ∪ (A × C)
4. (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D)
5. (A × B) ∪ (C × D) ⊂ (A ∪ C) × (B ∪ D)

###### Cartesian Product of Two Sets

1. Suppose A × B ⊂ C × D. Let a ∈ A and b ∈ B. Then (a, b) ∈ A × B, and so (a, b) ∈ C × D, by hypothesis. Thus a ∈ C and b ∈ D. Since a and b were arbitrary, A ⊂ C and B ⊂ D. The converse is left as an exercise.

2. We have (a, b) ∈ A × (B ∩ C) ⇔ a ∈ A and b ∈ B ∩ C.

⇔ a ∈ A and b ∈ B and b ∈ C.
⇔ (a, b) ∈ A × B and (a, b) ∈ A × C.
⇔ (a, b) ∈ (A × B) ∩ (A × C).

Property. does not say that every subset of C × D has the form A × B.

For example, if C = D = {1, 2}, then S = {(1, 1),(2, 2)} ⊂ C2, but S 6= A × B for any A, B.

Warning. Notice that property 5. is not equality.

Cartesian Product of Two Sets

For example, if A = [1, 3], B = [2, 5], C = [2, 4], D = [4, 6] (closed intervals in R), then A ∪ C = [1, 4] and B ∪ D = [2, 6].

The sets A × B, C × D, and (A ∪ C) × (B ∪ D) are all rectangles in
R2, and (A × B) ∪ (C × D) 6= (A ∪ C) × (B ∪ D) by inspection.

###### Cartesian Product of Two Sets. 