Examining mathematical claims, looking into definitions, formulating conjectures, attempting proofs, and presenting findings are all practiced by students in Exploring Mathematics, which also engages them with projects, activities, and examples that catch their attention. This text, which has few pre-requisites, is suitable for independent readers who desire an accessible introduction to theoretical mathematics as well as college students in their first and second years of study. The fundamental concepts of proving procedures, sets, functions, relations, and cardinality are included, along with a few extra topics that offer a wealth of opportunities for continued study. Students build intriguing examples and theorems using a problem-based approach to explore the content through a variety of exercises and projects. Students are assisted in exploring and mastering the subjects covered through in-text activities with detailed solutions or substantial hints in an appendix. Students have the chance to check their comprehension of the fundamental principles, pick up new ideas, and refine their mathematical creativity through the end-of-chapter tasks and projects.