D-Math-W05-Ch1 - SHEN'S CLASS NOTES-191 Class Notes for Discrete Structures I CS 191 Jan 10 2005 Textbook Discrete Mathematics 6/E By Richard

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SHEN’S CLASS NOTES-191 1 Class Notes for Discrete Structures I CS 191 Jan. 10, 2005 Textbook: Discrete Mathematics 6/E By Richard Johnsonbaugh
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SHEN’S CLASS NOTES-191 2 Chapter One Logic and Proofs Logic is the study of reasoning. It provides reasoning rules by which a new result (statement) can be obtained from known results and axioms. 1.1 Propositions Definition 1 A proposition is a declarative sentence that is either true or false, but not both. Example 1 The following sentences are propositions (a) The only positive integers that divide 7 are 1 and 7. (b) 1 + 1 = 3 (c) Earth is the only planet that contains life. Example 2 The following sentences are not propositions: (a) Please wear seat belt whenever you are driving. (b) Have a nice day!
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SHEN’S CLASS NOTES-191 3 We often use letters p , q , r , s , …, etc. to represent propositions. Example 3 p : 1+1 = 3. (We use p to represent the proposition 1+1 = 3.) The value of a proposition is called a truth value, which can be T (true) or F (false). Obviously, the truth value of p in the above example is F. Definition 2 A compound proposition is formed from existing propositions using logical operations. Commonly used logical operations are ( and operation), ( or operation), ¬ ( not operation), ( implication operation), ( bidirectional operation). We will explain them one by one.
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SHEN’S CLASS NOTES-191 4 (1) Let p and q be two propositions. p q is a compound proposition read as p and q . p q is also called the conjunction of p and q . is a binary operator called and operator. The truth value of p q depends on the truth values of p and q . We often use a truth table to show how to determine the truth value of p q from the truth values of p and q . The truth table for p q is shown below. Table 1 The truth table for conjunction, ( p and q ) p q p q T T T T F F F T F F F F As we can see that p q is true if and only if both p and q are true.
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SHEN’S CLASS NOTES-191 5 Example 4 p : It is raining q : It is cold. p q : It is raining and it is cold. p q is true if and only if both events are true: it is raining and it is cold. (2) Let p and q be two propositions. p q is a compound proposition read as p or q . p q is also called the disjunction of p and q . is a binary operator called or operator. The truth table for p q is as follows. Table 2 The truth table for disjunction, ( p or q ) p q p q T T T T F T F T T F F F Obviously, p q is true if and only if either p or q is true.
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SHEN’S CLASS NOTES-191 6 Note . p q will also be true if both p and q are true. Therefore, is also called inclusive or . Example 5 p : It is raining q : It is cold. p
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This note was uploaded on 04/12/2008 for the course CS 191 taught by Professor Shen during the Fall '06 term at University of Missouri-Kansas City .

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D-Math-W05-Ch1 - SHEN'S CLASS NOTES-191 Class Notes for Discrete Structures I CS 191 Jan 10 2005 Textbook Discrete Mathematics 6/E By Richard

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