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Discrete Math

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Lecture 1. Logic. Propositions.
The rules of logic specify the precise meaning of mathematical statements. For instance, the rules give us the meaning of such statements as, “There exists an integer that is greater than 100 that is a power of 2”, and “For every integer n the sum of the positive integers not exceeding n is ”. Logic is the basis of all mathematical reasoning, and it has practical applications to the design of computing machines, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science.
A proposition is a statement that is either true or false, but not both.
Letters are used to denote propositions, just as letters are used to denote variables. The conventional letters used for this purpose are p, q, r, s, … The truth value of a proposition is true, denoted by T, if it is a true proposition and false, denoted by F, if it is a false proposition.
We now turn our attention to methods for producing new propositions from those that we already have. Many mathematical statements are constructed by combining one or more propositions. New propositions, called compound propositions, are formed from existing propositions using logical operators.
Let p be a proposition. The statement “It is not the case that p” is another proposition, called the negation of p. The negation of p is denoted by p. The proposition p is read “not p”.
A truth table displays the relationships between the truth values of propositions. Table 1. The truth table for the negation of a proposition | P | p | TF | FT |
The negation of a proposition can also be considered the result of the operation of the negation operator on a proposition. The negation operator constructs a new proposition from a single existing proposition. We will now introduce the logical operators that are used to form new propositions from two or more existing...

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