For the binary operation multiplication modulo 5 ( _5) defined on… - Vidyadip

Question

For the binary operation multiplication modulo $5 (\times _5)$ defined on the set $S = \{1, 2, 3, 4\}.$ Write the value of $(3 \times _5 4^{-1})^{−1}$

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Answer

The composition table for $\times _5$ on the set $S = {1, 2, 3, 4}$ is

$\times _5$	$1$	$2$	$3$	$4$
$1$	$1$	$2$	$3$	$4$
$2$	$2$	$4$	$1$	$3$
$3$	$3$	$1$	$4$	$2$
$4$	$4$	$3$	$2$	$1$

Now,
$(3 \times _5 4^{-1})^{-1} = (3 \times _5 4)^{-1} [\because 4^{-1} = 4]$
$= 2^{-1} [3 \times _5 4 = 2]$
$= 3 [\because 2^{-1} = 3]$

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