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Cartesian Product of Sets

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description

  • Number of Elements in the Cartesian Product of Two Finite Sets
  • Cartesian Product of set of the Reals with Itself

definition

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p,q) : p  ∈ P, q  ∈ Q } If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = Ø

notes

Suppose A is a set of 2 colours and B is a set of  3 objects, i.e., A = {red, blue}and B = {b, c, s}, where b, c and s represent a particular bag, coat and shirt, respectively.
From the illustration given above we note that A × B = {(red,b), (red,c), (red,s), (blue,b), (blue,c), (blue,s)}. 
Note:
(i) Two ordered pairs are equal, if and only if  the corresponding first elements are equal and the second  elements are also equal.
(ii) If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e.,  if n(A) = p and n(B) = q,  then n(A × B) = pq. 
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B. 
(iv) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

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