Introduction:
The lac operon is a well-known example of an inducible operon in bacteria. It consists of three structural genes, lacZ, lacY, and lacA, and an adjacent regulatory region. The lac operon allows the bacterium to utilize lactose as a source of energy when glucose is scarce.

Explanation:
b) Lac operon is subject to catabolite repression:
Catabolite repression is a regulatory mechanism that controls the expression of certain operons in response to the presence of glucose. When glucose is present in the cell, the lac operon is repressed, and transcription is greatly reduced. This is because the lac operon is subject to negative control by the catabolite activator protein (CAP), which requires cyclic AMP (cAMP) for activation. When glucose is present, the intracellular levels of cAMP are low, and CAP cannot bind to the lac operon promoter. Therefore, the lac operon is repressed in the presence of glucose.

d) LacZ, lacY, lacA gene products are synthesized at low levels in the ininduced state:
When the lac operon is not induced, i.e., in the absence of lactose, the lacZ, lacY, and lacA genes are transcribed at a low level. This basal level of transcription allows for the synthesis of small amounts of the gene products required for lactose metabolism. However, the lac operon is not fully activated, and the levels of lacZ, lacY, and lacA gene products are relatively low.

a) The repressor binds to the promoter element:
This statement is not true. The lac operon is subject to negative control by the lac repressor protein. The lac repressor binds to the operator region of the lac operon, not the promoter element. Binding of the lac repressor to the operator prevents RNA polymerase from binding to the promoter and initiating transcription.

c) The promoter element is not essential for operon transcription:
This statement is not true. The promoter element is essential for operon transcription. The promoter is a DNA sequence that is recognized by RNA polymerase and is necessary for the initiation of transcription. In the case of the lac operon, RNA polymerase binds to the lac operon promoter and initiates transcription of the lacZ, lacY, and lacA genes when lactose is present and the lac repressor is not bound to the operator.

In conclusion, the true statements regarding the lac operon are that it is subject to catabolite repression (b) and that the lacZ, lacY, and lacA gene products are synthesized at low levels in the uninduced state (d). The false statements are that the repressor binds to the promoter element (a) and that the promoter element is not essential for operon transcription (c).