Relations between Steps

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A calculation with a number of steps, in which each step relates one expression to the next by equality, establishes that all the expressions are equal. However, it is usually the equality between the first and last expressions that is important to us. As mentioned earlier, the conclusion of the above four-step calculation is the equality between the first and last lines, that is, (n+1) 2 - n2 = 2xn + 1 . Formally, equality is a transitive relation. That is, if R = S and S = T, it is also the case that R = T. Two other relations that are transitive are the less-than relation, denoted by '<', and the at-most relation, denoted by X' 2 - Sometimes, our calculations use these

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2 It is strongly recommended that you pronounce the symbol X' as 'at most' rather than the more cumbersome 'less than or equal to'. Similarly, we recommend that you pronounce *>' as 'at least' rather than 'greater than or equal to'.

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3.3 Formatting Calculations

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relations in successive steps. Here, for example, is a calculation that constructs a rough estimate of the difference between 256 and 367: 367-256

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400-256

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367 < 400 } 200 < 256 }

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400 - 200

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The conclusion of this calculation is that 367 - 2 56 < 200. It illustrates the socalled conjunctional use of the less-than operator. In general, R<S <T means R < S and S <T. Transitivity of less-than means that a consequence of R < S < T is that R < T. Different relations may be combined in the same proof, but then there should be a logical justification for doing so. For instance, one step of a proof may assert, say, R < S, whereas the next asserts S ^ T. The inference is then that R < T. All such steps can be combined with equality steps, as in the last line above. However, it would be nonsense to combine '<' with '>' or >' in one calculation since then no inference can be made of the relation between the first and last expressions. The type of the expressions is arbitrary. They may denote real values, integer values, sets, relations, etc. In each case, the familiar equality symbol, '=', is used to denote equality of values. In particular, if R, S and T denote boolean values, we still use the equality symbol to denote equality. For example, a step in a proof might be = { E<F+l Here we are using the fact that the statement E^F is the same as the statement E<F+l whenever E and F denote integer values. In other words, the value of E^F (which is either true or false) is equal to the value of E<F+l. In in-line expressions, we often use the symbol '=' to denote equality of boolean values. One reason for this is to avoid ambiguity. For example, we write = E<F+l in order to avoid confusion with E^F = E <F+l E and F denote integer values. Property of integer arithmetic }

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3: Calculational Proof

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which means E < F and F = E and E < F+l. (So equality here means equality of integer values rather than equality of boolean values.) There is another reason for having two notations for equality of boolean values. This second reason, which is much more important than the first, is discussed in 5. Generally, in such calculations, the connecting relations will have lower precedence than the operators in the expressions they connect, this convention superseding any other precedence conventions. You should have no problem with this convention since it is clearly what is suggested by the layout.

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