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Tbe problem of transmission of separate messages to each of two receivers over a general binary-input broadcast channel is investigated. A new approach to a class of information-theoretic problems is developed and applied to obtain bounds on the cardinalities of auxiliary random variables. These bounds permit the calculation of two different regions of achievable rate pairs which are derived from the Cover-van der Meulen region of achievable rate triples. Numerical evaluation of these regions of rate pairs for two examples demonstrates that the region can be enlarged. This enlargement is accomplished by making internally consistent, as the true capacity region must be. The results display complex interactions between common and separate information in broadcast problems.