Prove that : (i) [latex]a b+c a+b+c[/latex] = 0 - Vidyadip

Question

Prove that : (i) [latex]\bar{a} \bar{b}+\bar{c} \bar{a}+\bar{b}+\bar{c}[/latex] = 0

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Answer

$\begin{aligned} & {[\bar{a} \bar{b}+\bar{c} \bar{a}+\bar{b}+\bar{c}]} \\ & =\bar{a} \cdot[(\bar{b}+\bar{c}) \times(\bar{a}+\bar{b}+\bar{c})] \\ & =\bar{a} \cdot(\bar{b} \times \bar{a}+\bar{b} \times \bar{b}+\bar{b} \times \bar{c}+\bar{c} \times \bar{a}+\bar{c} \times \bar{b}+\bar{c} \times \bar{c})\end{aligned}$

$\begin{aligned} & =\bar{a} \cdot(\bar{b} \times \bar{a})+\bar{a} \cdot(\bar{b} \times \bar{b})+\bar{a} \cdot(\bar{b} \times \bar{c})+\bar{a} \cdot(\bar{c} \times \bar{a})+ \\ & \bar{a} \cdot(\bar{c} \times \bar{b})+\bar{a} \cdot(\bar{c} \times \bar{c}) \\ & \end{aligned}$

$=0+0+\bar{a} \cdot(\bar{b} \times \bar{c})+0-\bar{a} \cdot(\bar{b} \times \bar{c})+0$

$=0$.

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