The author presents a new approach to understanding the underlying mathematical structure of Shannon's information measures, which provides answers to the following two questions for any finite number of random variables. (1) For any information-theoretic identity, is there a corresponding set-theoretic identity via the formal substitution of symbols? (2) For any set-theoretic identity, is there a corresponding information-theoretic identity and, if so, in what sense? The author establishes the analogy between information theory and set theory. Therefore, each information-theoretic operation can formally be viewed as a set-theoretic operation and vice versa. This point of view, which the author believes is of fundamental importance has apparently been overlooked in the past by information theorists. As a consequence the I-diagram, which is a geometrical representation of the relationship among the information measures, is introduced. The I-diagram is analogous to the Venn diagram in set theory. The use of the I-diagram is discussed.