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Unformatted text preview: 1 Negations 1.1 Welcome Welcome to the eighth module in our course on proofs, Negations. Before going on, please do the assigned readings. 1.2 Introduction You will frequently encounter the negation of statement A . The negation of the statement, is the statement NOT A , not surprisingly. Because statements cannot be both true and false, exactly one of A and NOT A can be true. 1.3 Negating Statements In some instances, finding the negation of a statement is easy. For example, if A is the statement A : f ( x ) has a real root. its negation NOT A is the statement NOT A : f ( x ) does not have a real root. When A already contains a NOT we use the rule that two negatives is a positive, or that one NOT cancels another NOT . For example A : 7 is not a divisor of 28. NOT A : 7 is a divisor of 28. A specific rule applies when negating a statement containing the word AND . For example, A : B AND C NOT A : NOT ( B AND C ) which is equivalent to ( NOT B ) OR ( NOT C ) Note that the connecting word has changed from AND to OR and that each term in the expression has been negated. The brackets are not needed because NOT precedes OR in logical evaluation, but the brackets are useful to emphasize the change. Here is a specific example. A : 4 T is isosceles and has perimeter 42. NOT A : 4 T is not isosceles or does not have perimeter 42. Similar to the conjunctive AND , a specific rule applies when negating a statement containing the word OR . A : B OR C NOT A : NOT ( B OR C ) which is equivalent to NOT B AND NOT C Note that the connecting word has changed from OR to AND and, again, each term in the expression has been negated. Again, the brackets are not needed because NOT precedes AND in logical evaluation, but the brackets are useful to emphasize the change. Here is a specific example A : 4 T is isosceles or it has perimeter 42. NOT A : 4 T is not isosceles and it does not have perimeter 42....
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This note was uploaded on 10/13/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.
- Spring '08