a The inducer for Lac operon of Escherichia coli is lactose (actually allolactose or metabolite of Lactose). This lac operon normally remains inactive. When lac operon contacts with lactose, the lactose acts as an inducer and combines with the repressor, and the repressor is detached from operator gene. Thus $RNA$ - repressor, and the repressor is detached from operator gertural genes and polymerase enzyme gets its passage and reaches to the structural starts the transcription.
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$A$ : Operator gene is functional when it is not blocked by repressor.
$R$ : Regulator gene produces active protein only which acts on operon system in $E$.coli