Question

Let, \[n(A) = n\]. Then find the number of all relations on A.
A.\[{2^n}\]
B. \[{2^{n!}}\]
C. \[{2^{{n^2}}}\]
D. None

Answer 1

Hint: Recall the total number of relations from a set A to another set B with\[n(A) = p\] and \[n(B) = q\]. Then substitute n for p and n for q in the obtained expression to get the required result.

Complete step by step solution: Complete step by step solution
The total number of relations from a set A to another set B, with \[n(A) = p\] and \[n(B) = q\]is \[{2^{pq}}\] .
Substitute n for p and n for q in the expression \[{2^{pq}}\] to obtain the required result.
Hence, \[{2^{n \times n}} = {2^{{n^2}}}\]

Option ‘C’ is correct

Additional Information: Relation is defined the relationship between two sets. There are 8 types of relation.
Empty relation: An empty relation is a relation when there is no relation between two sets.
Universal relation: An universal relation is a relation such that all elements of a set are related to every element of the set.
Identity relation: An identity relation is a relation such that all elements of a set are related to itself only.
Inverse relation: The inverse relation is a relation such that the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function.
Reflexive relation: A relation is said to be reflexive on set A, if each element of a set is related to itself.
Symmetric relation: A relation is said to be symmetric if (a,b) belongs to R implies (b,a) belongs to R.
Transitive relation: A relation is said to be transitive if (a,b), (b,c) belongs to R implies (a,c) belongs to R.

Note: Sometimes student get confused and substitute n for \[pq\] as it is given that \[n(A) = n\], and get the answer \[{2^n}\] but relation is defined from a set to another set, so here two sets are A and A, so we have to substitute n for p and n for q to obtain the required answer \[{2^{{n^2}}}\].