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Now we can use (5.4) and (5.15) to reduce the number of occurrences of 'p' and '<2' to at most one (possibly negated). In this particular example we obtain true = p = false = r . Finally, we use (5.4) and (5.15) again. The result is that the original formula is simplified to
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5: Calculational Logic: Part 1
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Just as before, this process can be compared with the simplification of an arithmetic expression involving continued addition, where now negative terms may also appear. The expression is simplified to
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by counting all the occurrences of p, q and r, an occurrence of -p cancelling out an occurrence of p. Again the details are different but the process is essentially identical. The two laws (5.4) and (5.15) are all that is needed to define the way that negation interacts with equivalence; using these two laws we can derive several other laws. A simple example of how these two laws are combined is a proof that -> false = true: --false = { false = false { true . Let us now see how associativity of equivalence is used in a simple calculation. We investigate the expression -<(p = q) to see whether it is possible to distribute negation through equivalence: { p = q = false { the law ^p = p = false with p := q } the law ~^p = p = fa\se with p := (p = q) } law true = p = p with p := false } law ->p = p = false with p := false }
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p = -^q . We have thus proved
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[Inequivalence] ->(p = q) = p = -^q . (5.16) Note how associativity of equivalence has been used silently in this calculation. Note also how associativity of equivalence in the summary of the calculation gives us two properties for the price of one. The first is the one proved directly:
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^(P = q) = (p = ^q) , the second comes free with associativity:
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5.6 Negation
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The proposition -*(p = q) is usually written p q. The operator is called inequivalence (or exclusive-or, abbreviated xor). As a final worked example, we show that inequivalence associates with equivalence:
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{ =Y
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~i(p = q) = P = -^ I
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->(p=q) = Y
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using symmetry of equivalence, the law (5.16) is applied in the form -^(p = q) = ->q = p with p,q := q,r }
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Exercise 5.17. Simplify the following. (Note that in each case it does not matter in which order you evaluate the subexpressions. Also, rearranging the variables and/or constants does not make any difference.) (a) false false ^ false . (b) true true true ^ true . (c) false true false true .
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(d) p = p = -ip = p = -^p .
(e) p q = q = p (f)p q =r =p (g) p = p ->p (h) p = p ->p
. . p = ~^p . p = -ip ->p .
Exercise 5.18. Using only equivalences and/or inequivalences, formalize the following statements. (a) None or both of p and q is true. (b) Exactly one of p and q is true. (c) Zero, two, or four of p, q, r and 5 is true. (d) One or three of p, q, r and 5 is true. Exercise 5.19. Prove that -> true = false. D D
5: Calculational Logic: Part 1 Exercise 5.20 (Double Negation). Prove the rule of double negation
-.-.p = p .
Exercise 5.21. The proof that inequivalence and equivalence associate with each other is summarized in the law any parenthesization being allowed, hi addition, any rearrangement of the variables is allowed because both equivalence and inequivalence are symmetric. Use these observations to list as many individual properties of equivalence and inequivalence as you can. In particular, deduce that inequivalence is associative.
Exercise 5.22 (Encryption). The fact that inequivalence is associative, that is
(p (q_ r}) = ((p q) r) ,
is used to encrypt data. To encrypt a single bit b of data, a key a is chosen and the encrypted form of b that is transmitted is a ^ b. The receiver decrypts the received bit, c, using the same operation4. That is, the receiver uses the same key a to compute a c. Show that, if bit b is encrypted and then decrypted in this way, the result is b independently of the key a. D Exercise 5.23. Let us return to the island of knights and knaves. In this question, there are two natives, A and B. Now, A says, 'B is a knight is the same as I am a knave'. What can you determine about A and B D Exercise 5.24. On the island of knights and knaves, you encounter two natives, A and B. What question should you ask A to determine whether A and B are different types D
5.7 Summary
In this chapter, we have explored the basic properties of equivalence (the equality of boolean values), negation and inequivalence. Equivalence is highly unusual in that it is reflexive, symmetric, transitive and associative. Inequivalence is symmetric and also associative. We have shown how these properties are exploited in a variety of situations and we have begun the introduction of an axiomatization of the logical connectives. See the appendix for a list of properties that have been established.
4 This operation is usually called 'exclusive-or' in texts on data encryption; it is not commonly known that exclusive-or and inequivalence are the same. Inequivalence can be replaced by equivalence in the encryption and decryption process. But, very few scientists and engineers are aware of the algebraic properties of equivalence, and this possibility is never exploited!
5.7 Summary