Question

how many binary relations are there on a set S with 9 distinct elements?​

Answer 1

Answer:

281

Step-by-step explanation:

a relation on s is defined as s×s. There are 9^2 number of ordered pair in reaction

Answer 2

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

DEFINITION TO BE MEMORISED

CARTESIAN PRODUCT OF SETS

Let A & B be any two non empty sets. The Cartesian product of A and B is denoted by A × B and defined by :

$\sf{A \times B = \{ \:(a,b) : a \in \: A \: , b \in \:B \}}$

BINARY RELATION

Let A and B be any two non empty sets. A binary relation R between A and B is a subset of A × B

NUMBER OF BINARY RELATIONS

If A and B be any two non empty sets such that

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

Then the number of binary relations between A & B

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

Using the same we can say that the number of binary relations on a set containing n is

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

TO DETERMINE

The number of binary relations are there on a set S with 9 distinct elements

CALCULATION

$\sf{Here \: \: n(S) = 9}$

Hence The number of binary relations the set S

$\sf{ \: = {2}^{(9 \times 9)} \: }$

$\sf{ = {2}^{81} \: }$

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