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How many Boolean functions in one variable are $\text{NOT}$ idempotent: i.e. they do not satisfy $\forall a.f(f(a)) = f(a).$

  1. $4$
  2. Infinite
  3. $1$
  4. $0$
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an idempotent operation is one that has no additional effect if it is called more than once with the same input parameters. ... In mathematics, an idempotent operation is one where $f(f(x)) = f(x)$

Possible functions $= 2^2 = 4$

$f(0)=0, f(1)=0$
$F(0)=0, f(1)=1  $
$F(0)=1, f(1)=0$ // NOT IDEMPOTENT 
$f(0)=1, f(1)=1$


if $a=0$
$f(f(a)) = f(a)$
$f(f(0)) = f(0)$

$1.$
$if f(0)=0$ 
$f(0) = f(0)$ // idempotent

$2.$ 
if $f(0)=1$
then $f(1)=f(0)$ Here we have assumed that $f(0)=1$ and for this function being NOT idempotent $f(1)$ will be $f(1)=0$

Therefore, (C) is the correct option.

Answer:

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