Question

If Number of Elements in Set A is 'p' and Number of Elements in Set B is 'q' then Prove that ,

(1) The Number of Possible relations from A to B is $\sf{{2}^{pq}}$

(2) The Number of Possible Reflexive Relations on A = $\sf{{2}^{p(p-1)}}$

- Copied and Irrelevant answers will be reported .



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Answer 1

Answer:

Total number of reflexive relations in a set with n elements = 2n Therefore, total number of reflexive relations set with 4 elements = 24.

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

Cartesian Product

For any two sets A & B the Cartesian product is denoted by A × B and defined as :

$\sf{ \: A \times B = \{ \: (x,y) : x \in \: A \: ,\: y \in \: B \}\: }$

Number of Subsets of a Set

Let A be a set with n elements

Then the Number of Subsets of the Set A is

$\sf \: { \: {2}^{ n} \: }$

GIVEN

If Number of Elements in Set A is p and Number of Elements in Set B is q

TO DETERMINE

(1) The Number of Possible relations from A to B is

$\sf{ \: {2}^{pq} }$

(2) The Number of Possible Reflexive Relations on A

$\sf{ \: {2}^{p(p - 1)} }$

EVALUATION

1.

By the given condition

$\sf{n(A) = p \: \: and \: \: n(B) = q}$

So

$\sf{ \: n(A \times B) = n(A) \times n(B) = pq }$

Now a relation from A to B is a subset of A × B

Since A × B has pq elements

So The Number of Possible relations from A to B is

$= \sf{ {2}^{pq} \: }$

2.

Again relation R on A is said to be Reflexive if

${ \: xRx \: \: holds \: \: x \in A \: }$

Now the number of elements in A is p

So The Number of Possible Reflexive Relations on A

$\sf \: { \: = {2}^{ {p}^{2} - p } \: }$

$\sf \: { \: = {2}^{{ p(p - 1)} } \: }$