# Rearrangement Operation as Replacement Operations

## Example of use of Replacement Operation

Let $\left({a, b, c, d}\right)$ be an ordered quadruple consisting of four variables whose values are to be rearranged into the order $\left({b, c, d, a}\right)$.

This can be implemented using replacement operations.

Let $t$ be a new variable which has been established for this purpose.

Then the sequence of replacement operations:

$t \gets a$
$a \gets b$
$b \gets c$
$c \gets d$
$d \gets t$

## Proof

Observing the values of the variables after each replacement operation:

Operation $a$ $b$ $c$ $d$ $t$
$t \gets a$ $a$ $b$ $c$ $d$ $a$
$a \gets b$ $b$ $b$ $c$ $d$ $a$
$b \gets c$ $b$ $c$ $c$ $d$ $a$
$c \gets d$ $b$ $c$ $d$ $d$ $a$
$d \gets t$ $b$ $c$ $d$ $a$ $a$

Hence the result.

Notice how the sequence:

$a \gets b, b \gets c, c \gets d, d \gets a$

does not do the job.

This is because, when $d \gets a$ is performed, $a$ no longer contains its original value, and the resulting ordered quadruple is $\left({b, c, d, b}\right)$.

$\blacksquare$