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Relations and Functions

Cartesian Product of Sets

  • Let P and Q be two non-empty sets. The Cartesian product of sets P and Q is denoted by P × Q and it is defined as the set of all ordered pairs of elements from P and Q i.e.,
    P × Q = {(p, q): pP, qQ}

    • For e.g., The Cartesian product of sets A = {1, 2, 3} and B = {4, 5} is
      A × B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

  • Two ordered pairs are equal if and only if their corresponding first elements are equal and the second elements are also equal.

  • If n (A) = p and n (B) = q,then n (A × B) = pq

  • If A and B are non-empty sets and either A or B is an infinite set, then so is A × B.

  • A × A × A = {(a, b, c): a, b, c A}. Here, (a, b, c) is called an ordered triplet.

  • To understand the concept of Cartesian product of sets, let us look at the given video.

 

Solved Examples

Example 1:

Let A = {1, 9}, B = {2, 4, 10, 11} and C = {2, 4, 6, 10}.

Find A × (BC) and show that it equals (A × B) ∩ (A × C).

Solution:

BC = {2, 4, 10}

A × (BC) = {1, 9} × {2, 4, 10}

A × (BC) = {(1, 2), (1, 4), (1, 10), (9, 2), (9, 4), (9, 10} … (1)

Now, we have to show that A × (BC) equals (A × B) ∩ (A × C).

A × B = {(1, 2), (1, 4), (1, 10), (1, 11), (9, 2), (9, 4), (9, 10), (9, 11)}

A × C = {(1, 2), (1, 4), (1, 6), (1, 10), (9, 2), (9, 4), (9, 6), (9, 10)}

(A × B) ∩ (A × C) = {(1,2), (1,4), (1,10), (9,2), (9,4), (9,10} … (2)

From equations (1) and (2...

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