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Ordered pair

A pair of objects, listed in a specific order, is called an ordered pair, for example (a, b) is an ordered pair of two elements a and b, a is called the first element and b, the second element. Two ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d.

Cartesian product of two sets

If A and B are any two non-empty sets, then Cartesian product of A and B is defined as A × B = {(a, b) : a A and b B}.

 

Notes:
If AB, C are three sets, then
  • A × (B ∪ C) = (A × B) ∪ (A × C)
  • A × (B ∪ C) = (A × B) ∪ (A × C)
  • A × (B – C) = (A × B) – (A × C)
  • If A ⊆ B, then (A × C) ⊆ (B × C)
  • If A ⊆ B, then (A × B) ∪ (B × A) = A2
  • If A ⊆ B and C ⊆ D then A × C ⊆ B × D
  • If A ⊆ B, then A × A ⊆ (A × B) ∪ (B × A)
  • A × B = B × A if and only if A = B
  • A × (B′ ∪ C′)′= (A × B) ∪ (A × C)
  • A × (B′ ∪ C′)′ = (A × B) ∪ (A × C)
  • (A × B) ∪ (S × T) = (A ∪ S) × (B ∪ T), where S and T are two sets
  • If A and B are two non-empty sets having n elements in common, then A × B and B × A have n2 elements in common




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