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By Alan Parker

The formulation of (1.3), (1.4), (1.5),(1.6) are all wrong!

Can't think it, 4 formulation in first web page are all have blunders. perhaps the writer fell so sleepy whilst he is scripting this booklet!

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7. For this case let us assume that the addition/subtraction operation has been set up as a pipeline. The first statement in the pseudo-code will cause the inputs x and 3 to be input to the pipeline for subtraction. After the first stage of the pipeline is complete, however, the next operation is unknown. In this case, the result of the first statement must be established. To determine the next operation the first operation must be allowed to proceed through the pipe. After its completion the next operation will be determined.

For these functions the asymptotic behavior is exhibited when n ≥ 1050. Although f1 ∈ Θ (en) it has a value of 1 for n < 1050. In a pragmatic sense it would be desirable to have a problem with time complexity f1 rather than f2. Typically, however, this phenomenon will not appear and generally one might assume that it is better to have an algorithm which is Θ (1) rather than Θ (en). One should always remember that the constants of order can be significant in real problems. 2 Induction Simple induction is a two step process: • Establish the result for the case N = 1 • Show that if is true for the case N = n then it is true for the case N = n+1 This will establish the result for all n > 1.

In this method it is required to demonstrate that a specific property cannot hold for a positive integer. 5 Induction 1. Let P (k) = TRUE denote that a property holds for the value of k. Also assume that P(0) does not hold so P(0) = FALSE. Let S be the set that From the well-ordering principle it is true that if S is not empty then S has a smallest member. Let j be such a member: 2. Prove that P(j) implies P(j-1) and this will lead to a contradiction since P(0) is FALSE and j was assumed to be minimal so that S must be empty.

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