logical operators and compound propositions, Schemes and Mind Maps of Mathematical logic

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Module 1. Lesson 2. Logical Operators and Compound Propositions | hope that you already have a clear understanding of the definition of a proposition because all other topics that we are going to discuss in Unit 1 are associated with propositions. Now, you are going to learn about the different methods of producing new propositions from given propositions. These new propositions, called compound propositions, are formed by combining one or more existing propositions using logical operators. We will discuss different logical operators, such as negation, conjunction, disjunction, implication, and bi-implication. Definition 2. Negation of a Proposition Let p be a proposition. The negation of p, denoted by ~p, is the proposition “It is not the case that p.” The notation ~p is read as “not p”. The truth value of ~p is the opposite of the truth value of p. Remark: The notation for negation operator is not standardized. You may also see other notations for negation operator such as ap, B,—p,p, Np, and ! p. Let us apply the concepts found in Definition 2 by answering Example 2 below. Example 2. Consider propositions p and q taken from the answer in Example 1. Express ~p and ~q in English sentences and determine their truth values. p: Manila is the capital of the Philippines. -T gq 5-4=2 -F Solution. According to Definition 2, the notation ~p is read as “not p” and it denotes the negation of the proposition p. ~p is the proposition “It is not the case that Manila is the capital of the Philippines.” Or we may write it simply as, “Manila is not the capital of the Philippines.” The truth value of ~p is false (F), which is just the opposite of the truth value of the given proposition p (true or T). On the other hand, ~q is read as “not q” and it denotes the negation of the proposition q.~¢q is the proposition “It is not the case that 5 — 4 = 2.” Or we may write it simply as, “5 — 4 ¢ 2”. The truth value of ~g is true (T), which is just the opposite of the truth value of the proposition q (false or F). LOGIC AND SET THEORY CYRENE A. CASPE / LEYTE NORMAL UNIVERSITY