Prove that $[(\to a + \to b) (\to b + \to c) (\to c + \to a)] = 2 [\to a \to b \to c]$ .

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Solution

$[¯ a + ¯ b ¯ b + ¯ c ¯ c + ¯ a]$
$\Rightarrow (¯ a + ¯ b) . [(¯ b + ¯ c) \times (\to c + \to a)]$
$(¯ a + ¯ b) . [¯ b \times \to c + \to b \times \to a + \to c \times \to c + \to c \times \to a]$
$\Rightarrow \to a . \to b \times \to c + \to a . \to b \times \to a + \to a . \to c \times \to c + \to a . \to c \times \to a + \to b . \to b \times \to c + \to b . (\to b \times \to a) + \to b . \to c \times \to c + \to b (\to c \times \to a)$
$\Rightarrow \to a . \to b \times \to c + \to b . \to c \times \to a .$
$= 2 \to a . \to b \times \to c + \to b . a (b c z, \to a . \to b \times \to c = \to b . \to c \times \to a) .$
$\Rightarrow 2 [\to a \to b \to c]$

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Prove that [(→a+→b) (→b+→c) (→c+→a)]=2[→a→b→c].

Prove that $[(\to a + \to b) (\to b + \to c) (\to c + \to a)] = 2 [\to a \to b \to c]$ .