Question

Let \(A=\\{1,2,3,4,5,6\\} .\) How many different relations are there on the set \(A\) ?

Short answer

The number of different relations on the set \(A\) is \(2^{36}\).

Step-by-step solution

Understanding the Problem

A relation from a set \(A\) to a set B is a subset of the cartesian product \(A \times B\). In this case, we are asked to find the number of relations on the set \(A = \{1,2,3,4,5,6\}\), which means finding the number of subsets in the cartesian product \(A \times A\).

Calculate the Cardinality of the Cartesian Product

The cardinality of the cartesian product \(A \times A\) is the product of the cardinalities of \(A\) and \(A\), which is \(6 \times 6 = 36\). That means the cartesian product \(A \times A\) contains 36 pairs.

Calculate the Number of Subsets

To find the number of subsets in a set of cardinality \(n\), we use the formula \(2^n\). In this case, the number of subsets in \(A \times A\) is \(2^{36}\)