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Maths-

General

Easy

Question

If a set $A$ contains $n$ elements, then which of the following cannot be the number of reflexive relations on the set $A ?$

(a) $2^{n}$
$2^{n - 1}$
$2^{n^{2} - 1}$
$2^{n + 1}$

The correct answer is: $2 to the power of n plus 1 end exponent$

A relation on set $A$ is a subset of $A cross times A$
Let $A equals open curly brackets a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma a subscript n end subscript close curly brackets.$ Then, $a$ reflexive relation on $A$ must contain at least $n$ elements $open parentheses a subscript 1 end subscript comma a subscript 1 end subscript close parentheses comma open parentheses a subscript 2 end subscript comma a subscript 2 end subscript close parentheses comma horizontal ellipsis comma left parenthesis a subscript n end subscript comma a subscript n end subscript right parenthesis$
$therefore$ Number of reflexive relations on $A$ is $2 to the power of n to the power of 2 end exponent minus n end exponent$
Clearly, $n to the power of 2 end exponent minus n equals n comma n to the power of 2 end exponent minus n equals n minus 1 comma n to the power of 2 end exponent minus n equals n to the power of 2 end exponent minus 1$ have solutions in $N$ but $n to the power of 2 end exponent minus n equals n plus 1$ is not solvable in $N.$
So, $2 to the power of n plus 1 end exponent$ cannot be the number of reflexive relations on $A$

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