Question

A 3-Input majority gate is defined by the logic function $M(a,b,c)=ab+bc+ca.$ Which one of the following gate is represented by the function $M\overline{\left(M\right(a,b,c)},M(a,b,\overline{c}),c)?$

• A. 3- Input NAND gate
• B. 3-Input XOR gate
• C. 3-Input NOR gate
• D. 3-Input XNOR gate

Answer 1

The correct option is B 3-Input XOR gate

$M(a,b,c)=ab+bc+ca$
$\overline{M(a,b,c)}=\overline{ab+bc+ca}$
$=\overline{ab}.\overline{bc}.\overline{ca}$
$=(\overline{a}+\overline{b})(\overline{b}+\overline{c})(\overline{c}+\overline{a})$
$M\left(M\right(a,b,\overline{c}),M(a,b,\overline{c}),c)=\left[\begin{array}{c}(\overline{ab}.\overline{bc}.\overline{ca})(ab+b\overline{c}+\overline{c}a)\\ +(ab+b\overline{c}+\overline{c}a)\left(c\right)+(\overline{ab.}.\overline{bc}.\overline{ca})c\end{array}\right]$
$=\left[\begin{array}{c}(\overline{a}+\overline{b})(\overline{b}+\overline{c})(\overline{c}+\overline{a})(ab+b\overline{c}+\overline{c}a)\\ +abc+(\overline{a}+\overline{b})(\overline{b}+\overline{c})(\overline{c}+\overline{a})c\end{array}\right]$
$=(\overline{a}+\overline{b})(\overline{b}+\overline{c})(\overline{c}+\overline{a})[ab+b\overline{c}+\overline{c}a+c]+abc$
$=(\overline{a}+\overline{b})(\overline{b}+\overline{c})(\overline{c}+\overline{a})[a+b+c]+abc$
$=(\overline{a}\overline{b}+\overline{b}\overline{c}+\overline{c}\overline{a})[a+b+c]+abc$
$F=a\overline{b}\overline{c}+b\overline{c}\overline{a}+c\overline{a}\overline{b}+abc$
$F=A⨁B⨁C$