By Franco Vivaldi
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Sample text
However, these rules are not as established as those for arithmetical operators, and we parentheses should be used whenever there is a potential ambiguity. A relational operator binds more tightly than any boolean operator, thereby removing any ambiguity. Thus the expression 1 > 2 ⇒ 2 > 1 can only be interpreted as (1 > 2) ⇒ (2 > 1), because 1 > (2 ⇒ 2) > 1 is syntactically incorrect (the operands of ⇒ must be boolean). Redundant parentheses may still be used to improve readability. 8) we see that the expression FALSE ⇒ Q is true for any value of Q.
1. BOOLEAN EXPRESSIONS 53 P implies Q Q follows from P P only if Q P is sufficient for Q. if P, then Q The statement P is the hypothesis, while Q is the conclusion. 1), for example), we shall study this operator thoroughly. As with arithmetical operators, the order of evaluation of boolean operators may be altered by means of parentheses (F ∧ T) ⇒ T F ∧ (T ⇒ T). The expressions above evaluate to TRUE and FALSE, respectively. In absence of parentheses, the implication operator binds more tightly than ∧ and ∨, while the latter have the same precedence.
We have seen that a two-element sequence may be called an (ordered) pair, and a three-element sequence a triple. Occasionally one sees the terms quadruple or quintuple (I wouldn’t go much beyond that), while an n-element sequence may be called an nuple. A finite sequence of numbers may be called a vector, in which case we speak of dimension rather than length. As an exercise in notation for sequences, let us define the n-dimensional sphere. Its elements are n-dimensional vectors, subject to a constraint.