e only most likely. But by suitably increasing the message sequence length n, one can make such likelihood arbitrarily high, and thus the probability of error can be made arbitrarily small. While this approach does not constitute a formal demonstration of the CCT (unlike the analysis that preceded), it provides a more intuitive and simple description thereof. Approach 2 This is similar to the first approach, but this time we look at the channel from the output end. The question we ask is: what is the condition for any output sequence to be distinguishable, i.e., to correspond to a unique (nonconfusable) fan of input sequences? Let us go through the same reasoning as in the previous approach. Referring again to Fig. 13.5, it is seen that a fan of 2nH(X|Y) most likely (and equiprobable) input sequences of the typical set of Xn corresponds to any output sequence yj of the typical set of Yn. Define M as the number of nonconfusable input fans, i.e., the fans that are attached to distinguishable output sequences. The number of input sequences corresponding to the M distinguishable outputs is, thus, M2nH(X|Y) Since there are 2nH(X) elements in the typical set Xn, we have M2nH(X|Y) < 2nH(X) and, hence, M (2n[H(X)-nH(XY) (13.35) 2nH(Y;X)=2nH(X;Y)2nC where we used the property of mutual information H(Y; X) = H(X; Y). The result in Eq. (13.35) leads to the same condition expressed in Eq. (13.34), i.e., R < C, which defines the channel capacity as the maximum rate for accurate transmission. Channel capacity and coding theorem Approach 3 The question we ask here is: can one construct a code for which the transmission error can be made arbitrarily small? We show that there is at least one such code. Assume a source X for which the channel capacity is achieved, i.e., H(X; Y) = C, and a channel rate R < C. We can define a code with a set of 2nR codewords (2nR<2nC), chosen successively at random from the typical set of 2nH(X) sequences, with a uniform probability. Assume that the codeword Xi is input to the channel and the output