Large Language Models (LLMs) have demonstrated impressive capabilities in generating diverse and contextually rich text. However, concerns regarding copyright infringement arise as LLMs may inadvertently produce copyrighted material. In this paper, we first investigate the effectiveness of watermarking LLMs as a deterrent against the generation of copyrighted texts. Through theoretical analysis and empirical evaluation, we demonstrate that incorporating watermarks into LLMs significantly reduces the likelihood of generating copyrighted content, thereby addressing a critical concern in the deployment of LLMs. Additionally, we explore the impact of watermarking on Membership Inference Attacks (MIAs), which aim to discern whether a sample was part of the pretraining dataset and may be used to detect copyright violations. Surprisingly, we find that watermarking adversely affects the success rate of MIAs, complicating the task of detecting copyrighted text in the pretraining dataset. Finally, we propose an adaptive technique to improve the success rate of a recent MIA under watermarking. Our findings underscore the importance of developing adaptive methods to study critical problems in LLMs with potential legal implications.