Question

Find the number of reflexive relations from set A to A, defied as \[A = a,b,c\].

Answer 1

Hint: First we have to find the total number of ways in which the elements can be arranged to form ordered pairs of reflexive relations. Then the total number of reflexive relations can be found out by the formula \[{2^m}\] where m is the total number of ordered pairs.

Complete step-by-step solution:
Before solving let’s see what the reflexive relation is. A binary relation R over a set A is said to be reflexive if each element of set A couples to itself. Mathematically,
 If \[(a,a) \in R\]\[\forall a \in A\].
 In other words, for every \[a \in A\], \[aRa\].
The given set \[A = a,b,c\].
Here the number of elements in A is 3. Let (p,q) be the ordered pairs of reflexive relation from \[A \to A\].
Now p can be chosen in 3 ways (i.e. a, b or c), q also can be chosen in 3 ways. So the number of ordered pairs will be\[3 \times 3 = 9\].
But according to the definition of reflexive relation, the ordered pairs in set R must include \[(x,x)\]. For e.g. the relations are as follows.
\[ {R_1} = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {1,2} \right)} \right\} \\
{R_2} = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {2,1} \right)} \right\} \\
{R_3} = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {1,2} \right),\left( {2,1} \right)} \right\} \\
  ........................... \\
  ........................... \\ \]
and so on.
 As the number of \[(x,x)\] is 3. Those are respectively \[\left( {1,1} \right){\text{ }}\left( {2,2} \right){\text{ }} and {\text{ }}\left( {3,3} \right)\] and then the number of ordered pair will be \[9 - 3 = 6\]
Hence the total number of reflexive relations is \[{2^6} = 64\].

Note: A binary relation from set P to Q is a subset of the Cartesian product \[P \times Q\]. In an alternative method, we can directly calculate the total number of reflexive relations by the following formula. The total number of reflexive relations of a set having ‘n’ number of elements is \[{2^{n(n - 1)}}\]. If there is at least one element such that \[(a,a) \notin R\] \[\forall a \in A\], then the relation can’t be said reflexive.