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as asserting the equality of all of x, y and z Or do we read it 'associatively' as
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or, equally, as
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in just the same way as we would read x+y+z The two readings are unfortunately not the same (for example, true = false = false is false according to the first reading but true according to the second and third readings). As we shall see, there are advantages in both readings and it is a major drawback to have to choose one in favour of the other. It would be very confusing and, indeed, dangerous to read x = y = z in any other way than x = y and y = z; otherwise, the meaning of a sequence of expressions separated by equality symbols would depend on the type of the expressions. Also, the conjunctional reading (for other types) is so universally accepted for good
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>.3 Examples of the Associativity of Equivalence
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reasons that it would be quite unacceptable to try to impose a different convention. The solution to this dilemma is to use two different symbols to denote equality of boolean values the symbol '=' when the transitivity of the equality relation is to be emphasized and the symbol 's' when its associativity is to be exploited. Accordingly, we will write both p = q and p == q. (As the reader will have observed, we have been doing this for some time now. It is only now, however, that wje have been able to provide a full explanation.) When p and q are expressions denoting boolean values, these both mean the same. But a continued expression
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comprising more than two boolean expressions connected by the '=' symbol, is to be evaluated associatively i.e. as (p = q} = r or p = (q = r), whichever is the most convenient whereas a continued expression
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is to be evaluated conjunctionalty i.e as p = q and q-r. More generally, a continued equality of the form
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means that all of pi, p z , . - . , pn are equal, whilst a continued equivalence of the form
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has the meaning given by fully parenthesizing the expression (in any way whatsoever, since the outcome is not affected) and then evaluating the expression as indicated by the chosen parenthesization. Note that when n is 2 we may use either symbol. That is, p = q and p = q have the same meaning. Moreover, we recommend that the '=' symbol is pronounced as 'equivales'; being an unfamiliar word, its use will help to avoid misunderstanding.
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5.3 Examples of the Associativity of Equivalence
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This section contains a number of examples illustrating the effectiveness of the associativity of equivalence. Even and Odd Numbers. The first example is particularly beautiful. It is the following property of the predicate even on numbers. (A number is even exactly when it is divisible by two.) m+n is even = m is even = n is even .
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5: Calculational Logic: Part 1 It will help if we refer to whether or not a number is even or odd as the parity of the number. Then, if we parenthesize the statement as m+n is even = (m is even = n is even) , it states that the number m+n is even exactly when the parities of m and n are both the same. Parenthesizing it as (m+n is even = m is even) = n is even , it states that the operation of adding a number n to a number m does not change the parity of m exactly when n is even. Another way of reading the statement is to use the fact that, in general, the equivalence p = q==r is true exactly when an odd number of p, q and r is true (see Exercise 5.2). So the property captures four different cases: or or or ((m+n is even) ((m+n is odd) ((m+n is odd) ((m+n is even) and and and and (m is even) (m is odd) (m is even) (m is odd) and and and and (n is even)) (n is even)) (n is odd)) (n is odd)) .
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The beauty of this example lies in the avoidance of case analysis. There are four distinct combinations of the two booleans 'm is even" and 'n is even'. Using the associativity of equivalence the value of 'm+n is even' is expressed in one simple formula, without any repetition of the component expressions, rather than as a list of different cases. Avoidance of case analysis is vital to effective reasoning. Exercise 5.3. The sign of a number says whether or not the number is positive. For non-zero numbers x and y, the product xxy is positive if the signs of x and y are equal. If the signs of x and y are different, the product xxy is negative. Assuming that x and y are non-zero, this rule is expressed as xxy is positive = x is positive = y is positive . Interpret the two different ways of parenthesizing the equivalences and enumerate the different properties of the sign of a number that this one equivalence captures. D Full Adder. A full adder is a component of a circuit to add two binary numerals. An addition unit comprises a chain of full adders, the number of full adders being equal to the bit length of the numerals to be added. Each full adder has three inputs and two outputs. The three inputs are two bits to be added and a carrier bit, the carrier bit being 'carried over' from previous additions in the chain. Let us suppose the bits to be added are a and b and the carrier bit is c. Let A, B and C be the propositions a-l,b = l and c = I . The output is a bit d, which is the least significant bit of a+b+c, and a new carrier bit, the most significant bit of a+b+c. Let us suppose that D is the proposition d = 1. It is easy to see that d = 1 exactly
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