Negation
Given any proposition p, another proposition, called the negation of p, can be formed by writing “It is not true that . . .” or “It is false that . . .” before p or, if possible, by inserting in p the word “not.” Symbolically, the negation of p, read “not p,” is denoted by ?p
The truth value of ?p depends on the truth value of p as follows:
Definition 4.3: If p is true, then ?p is false; and if p is false, then ?p is true.The truth value of ?p may be defined equivalently by the table in Fig. 4-1(c). Thus the truth value of the negation of p is always the opposite of the truth value of p.
EXAMPLE 4.3 Consider the following six statements:
(a1) Ice floats in water. (a2) It is false that ice floats in water. (a3) Ice does not float in water.
(b1) 2 + 2 =5 (b2) It is false that 2 + 2 = 5. (b3) 2 + 2 _= 5
Then (a2) and (a3) are each the negation of (a1); and (b2) and (b3) are each the negation of (b1). Since (a1) is true, (a2) and (a3) are false; and since (b1) is false, (b2) and (b3) are true.
Remark: The logical notation for the connectives “and,” “or,” and “not” is not completely standardized. For example, some texts use:
p & q, p ? q or pq for p ? q
PROPOSITIONS AND TRUTH TABLES
Let P(p, q, . . .) denote an expression constructed from logical variables p, q, . . ., which take on the value TRUE (T) or FALSE (F), and the logical connectives ?, ?, and ? (and others discussed subsequently). Such an expression P(p, q, . . .) will be called a proposition.
The main property of a proposition P(p, q, . . .) is that its truth value depends exclusively upon the truth values of its variables, that is, the truth value of a proposition is known once the truth value of each of its variables is known. A simple concise way to show this relationship is through a truth table. We describe a way to obtain such a truth table below.
Consider, for example, the proposition ?(p??q).
There is then a column for each “elementary” stage of the construction of the proposition, the truth value at each step being determined from the previous stages by the definitions of the connectives ?, ?, ?. Finally we obtain the truth value of the proposition, which appears in the last column.
Remark: In order to avoid an excessive number of parentheses, we sometimes adopt an order of precedence for the logical connectives. Specifically, ?has precedence over ? which has precedence over ?
For example, ?p ? q means (?p) ? q and not ?(p ? q).
TAUTOLOGIES AND CONTRADICTIONS
Some propositions P(p, q, . . .) contain only T in the last column of their truth tables or, in other words, they are true for any truth values of their variables. Such propositions are called tautologies. Analogously, a proposition P(p, q, . . .) is called a contradiction if it contains only F in the last column of its truth table or, in other words, if it is false for any truth values of its variables. For example, the proposition “p or not p,” that is, p ??p, is a tautology, and the proposition “p and not p,” that is, p??p, is a contradiction. This is verified by looking at their truth tables in Fig. 4-5. (The truth tables have only two rows since each proposition has only the one variable p.)
Note that the negation of a tautology is a contradiction since it is always false, and the negation of a contradiction is a tautology since it is always true.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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