Theorems on Cartesian product of sets

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)
(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A  A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

Theorem 5
If A is a subset of B, (A×C) is a subset of (B×C) for any set C.

Theorem 6
If A is a subset of B and C is a subset of D, (A×C) is a subset of (B×D)

Theorem 7
For any sets A,B,C , D,

(A×B) ∩(C×D) = (A∩C) ×(B∩D)

Theorem 8

For any three sets A,B,C

i. (A×(B’ U C’) = (A×B) ∩(A×C)
ii. (A×(B’ ∩ C’) = (A×B) U(A×C)

Theorem 9
When A and B are two non-empty sets having n elements in common, (A×B) and (B×A) will have n² elements in common.