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The complexity of an electronic switching circuit is defined in a sufficiently general way so that most definitions which are presently used may be included. If Â¿(p, q) is the complexity of a p input q output circuit which has been minimized then we may define E(p, q) as the maximum of Â¿(p, q) over all p input, q output circuits. In spite of the generality of the definition of complexity one may obtain the following inequality which gives upper and lower bounds on this maximum complexity: C12r/rÂ¿E(p, q)Â¿C22r/r where r = p+log2 q. In this expression C1 and C2 are constants independent of p and q which depend upon the definition of complexity. These theoretical bounds are compared with those obtained from a few known circuiit designs.