Cartesian Product and Binary relations
Cartesian Product and Binary relations will be explained with the help of example.
Example 1:-
Lets Consider two nonempty sets A and B
A = { 1 , 2 } and B = { 3 , 4 }
Cartesian Product:-
A x B = { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) }
Binary Relations:- $$R_1=\{\;\;\}\;\\\\R_2=\{\;(1,3)\;\}\\\\R_3=\{\;(1,4)\;\}\\\\R_4=\{\;(2,3)\;\}\\\;\\R_5=\{\;(2,4)\;\}$$
$$R_6=\{\;(1,3),(1,4)\;\}\\\\R_7=\{\;(1,3),(2,3)\;\}\\\\R_8=\{\;(1,3),(2,4)\;\}\;\\\\\\R_9=\{\;(1,4),(2,3)\;\}\\\\R_{10}=\{\;(1,4),(2,4)\;\}\;\\\\R_{11}=\{\;(2,3),(2,4)\;\}$$
$$R_{12}=\{\;(1,3),(2,3),(2,4)\;\}\\\\R_{13}=\{\;(1,3),(2,3),(2,4)\;\}\\\\R_{14}=\{\;(1,3),(1,4),(2,4)\;\}\\\\R_{15}=\{\;(1,4),(2,3),(2,4)\;\}\\\\R_{16}=\{\;(1,3),(1,4),(2,3),(2,4)\;\}\\\\$$
Lets checks the functions in the above relations
$$Only\;R_7\;,\;R_8\;,\;R_9\;and\;R_{10}\\are\;the\;functions.$$
Because there is no repetition in the first elements in the ordered pairs of the above mentioned relations and Domains of all the relations are equal to set A.
Mathematically written as
$$f\;:\;A\rightarrow B$$
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Definitions:-
Cartesian Product:-
Let A and B be two non empty sets, cartesian product is the set of all ordered pairs (x,y) such that x belongs to A and y belongs to B and it is denotes by A x B.
In set builder notation cartesian product can be defined as
$$A\times B\;=\;\{\;(x,y)\;\vert\;x\in A\;\wedge\;y\in B\;\}$$
Binary relation:-
Binary relation is the subset of the cartesian product.
Domain:-
Domain is the set of first elements of ordered pairs forming a relation.
Range:-
Range is the set of second elements of ordered pairs forming a relation.
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Example 2:-
Lets find A x A its relations and functions
A = { 1 , 2 } and again A = { 1 , 2 }
Cartesian Product:-
A x A = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) }
Binary relation:- $$R_1=\{\;\;\}\;\\\\R_2=\{\;(1,1)\;\}\\\\R_3=\{\;(1,2)\;\}\\\\R_4=\{\;(2,1)\;\}\\\;\\R_5=\{\;(2,2)\;\}$$
$$R_6=\{\;(1,1),(1,2)\;\}\\\\R_7=\{\;(1,1),(2,1)\;\}\\\\R_8=\{\;(1,1),(2,2)\;\}\;\\\\\\R_9=\{\;(1,2),(2,1)\;\}\\\\R_{10}=\{\;(1,2),(2,2)\;\}\;\\\\R_{11}=\{\;(2,1),(2,2)\;\}$$
$$R_{12}=\{\;(1,1),(1,2),(2,1)\;\}\\\\R_{13}=\{\;(1,1),(1,2),(2,2)\;\}\\\\R_{14}=\{\;(1,1),(2,1),(2,2)\;\}\\\\R_{15}=\{\;(1,2),(2,1),(2,2)\;\}\\\\R_{16}=\{\;(1,1),(1,2),(2,1),(2,2)\;\}\\\\$$
$$Again\;only\;R_7\;,\;R_8\;,\;R_9\;and\\R_{10}\;are\;the\;functions.$$
Because there is no repetition in the first elements in the ordered pairs of the above mentioned relations and Domains of all the relations are equal to set A.
Mathematically written as
$$f\;:\;A\rightarrow A$$
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