Question

Let $A=\{1,2,3\}$. The total number of distinct relations that can be defined over $A$ is:
A.$6$
B.$8$
C.${{2}^{9}}$
D.None of these

Answer 1

Hint: We have given set $A=\{1,2,3\}$ and we have to find the total number of distinct relations that can be defined over $A$. So we know that the total number of binary relations over the set $A$ will be ${{2}^{{{n}^{2}}}}$.

Complete step-by-step answer:
We have given set $A=\{1,2,3\}$ and we have to find the total number of distinct relations that can be defined over $A$. So we know that the total number of binary relations over the set $A$ will be ${{2}^{{{n}^{2}}}}$.
Now here $n=3$.
Total number of distinct relations over the set $A$$={{2}^{{{3}^{2}}}}$.
Simplifying in simple manner we get,
Total number of distinct relations over the set $A$$={{2}^{9}}$.
So we get the total number of distinct relations over set $A$ is ${{2}^{9}}$.
Therefore, the correct answer is option (C).

Additional information:
Sets, in mathematics, are an organized collection of objects and can be represented in set-builder form or roster form. Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set. The order of a set defines the number of elements a set is having. It describes the size of a set. The order of sets is also known as the cardinality. In set theory, the operations of the sets are carried when two or more sets combine to form a single set under some of the given conditions.

 Note: We have given set $A=\{1,2,3\}$. Here the concept of total number of binary relations over the set $A$ will be ${{2}^{{{n}^{2}}}}$ should be known. Remember this formula.