RELATIONS AND FUNCTIONS G . W. Leibnitz ( – ) RELATIONS AND FUNCTIONS brackets and grouped together in a particular order, i.e., ( p,q ), p ∈ P and q ∈ Q . This leads to the following definition: 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 = φ From the illustration given above we note that A × B = {(red, b ), (red, c ), (red, s ), (blue, b ), (blue, c ), (blue, s )}. Again, consider the two sets: A = {DL, MP, KA}, where DL, MP, KA represent Delhi, Madhya Pradesh and Karnataka, respectively and B = { , , }representing codes for the licence plates of vehicles issued by DL, MP and KA . If the three states, Delhi, Madhya Pradesh and Karnataka were making codes for the licence plates of vehicles, with the restriction that the code begins with an element from set A, which are the pairs available from these sets and how many such pairs will there be ( Fig . )?
The available pairs are:(DL, ), (DL, ), (DL, ), (MP, ), (MP, ), (MP, ), (KA, ), (KA, ), (KA, ) and the product of set A and set B is given by A × B = {(DL, ), (DL, ), (DL, ), (MP, ), (MP, ), (MP, ), (KA, ), (KA, ), (KA, )}. It can easily be seen that there will be such pairs in the Cartesian product, since there are elements in each of the sets A and B. This gives us possible codes. Also note that the order in which these elements are paired is crucial.
For example, the code (DL, ) will not be the same as the code ( , DL). As a final illustration, consider the two sets A= { a , a } and B = { b