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$3.(15$ points$)$ Design a majority circuit with three inputs, whose output is $1$ if the three input variables have more $1\text{'}$s than $0\text{'}$s$.$ For example, the output of $001$ is $0,$ and the output of $110$ is $1.$

$($a$)(5$ points$)$ Show the truth table.

$($b$)(5$ points$)$ Show a simplified Boolean equation of this circuit.

$($c$)(5$ points$)$ Draw a logic diagram of this circuit.