This is a definition that our professor mentioned in class $-$the process of turning on the expression of gene in the response to a substance in the enviornment is called induction.
From wikipedia $-$ in genetics and biochemistry the removal of repression, such as of an operon so that gene transcription occurs or is enhanced, with the net result frequently being elevation of the level of a specific enzyme. This effect is called derepression.
In lac operon when Allolactose molecule is present in the medium (in the bacterial cell) it binds to the monomer of repressor and reduces the binding affinity of repressor to the operator site and the RNA polymerase is free to synthesis a polycistronic mRNA. Thus the allolactose molecule turns on the expression of lac operon by removing repression.
So I conclude that in lac operon induction is same as derepression. But does it hold true for all operons or there are operons in which induction is achieved without repressing the repressor?