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Lecture 7 – Jan 28, 2002
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Chapter 2 The Logic of Quantified Statements
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Section 2.1 Predicates and Quantified Statements I
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Predicates A predicate is a sentence that contains a finite number of variables, and becomes a statement when values are substituted for the variables. “x flies like a y.” Let x be “time” and y be “arrow.” Let x be “fruit” and y be “banana.”
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Domains of Predicate Variables The domain D of a predicate variable x is the set of all values that x may take on. Let P(x) be the predicate. x is a free variable. The truth set of P(x) is the set of all values of x D for which P(x) is true.
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The Universal Quantifier The symbol is the universal quantifier. The statement x S, P(x) means “for all x in S, P(x),” where S D. x is a bound variable, bound by the quantifier . The statement is true if P(x) is true for all x in S. The statement is false if P(x) is false for at least one x in S.
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Examples Statement “7 is a prime number” is true. Predicate “x is a prime number” is neither true nor false. Statements “ x {2, 3, 5, 7}, x is a prime number” is true. “ x {2, 3, 6, 7}, x is a prime number” is false.
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Examples of Universal Statements x {1, …, 10}, x 2 > 0. x {1, …, 10}, x 2 > 100. x R, x 3 – x 0. x R, y R, x 2 + xy + y 2 0. x , x 2 > 100.
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The Existential Quantifier The symbol is the existential quantifier. The statement x S, P(x) means “there exists x in S such that P(x),” S D. x is a bound variable, bound by the quantifier . The statement is true if P(x) is true for at least one x in S. The statement is false if P(x) is false for all x in S.
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Examples of Universal Statements x {1, …, 10}, x 2 > 0. x {1, …, 10}, x 2 > 100. x R, x 3 – x 0. x R, y R, x 2 + xy + y 2 0. x , x 2 > 100.
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Negations of Universal Statements The negation of x S, P(x) is the statement x S, P(x). If “ x R, x 2 > 10” is false, then “ x R, x 2 10” is true.
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Negations of Existential Statements The negation of x S, P(x) is the statement x S, P(x). If “ x R, x 2 < 0” is false, then “ x R, x 2 0” is true.
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Example: Negation of a Universal Statement p = “Everybody likes me.” Express p as x {all people}, x likes me. p is the statement x {all people}, x does not like me. p = “Somebody does not like me.”
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Example: Negation of an Existential Statement p = “Somebody likes me.” Express p as x {all people}, x likes me. p is the statement x {all people}, x does not like me. p = “Everyone does not like me.” p = “Nobody likes me.”
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Lecture 8 – Jan 29, 2002
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Section 2.2 Predicates and Quantified Statements II
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Multiply Quantified Statements Multiple universal statements x S, y T, P(x, y) The order does not matter. Multiple existential statements x S, y T, P(x, y) The order does not matter.
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Multiply Quantified Statements Mixed universal and existential statements x S, y T, P(x, y) y T, x S, P(x, y) The order does matter. What is the difference? Compare x R, y R, x + y = 0. y R, x R, x + y = 0.
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Negation of Multiply Quantified Statements Negate the statement x R, y R, z R, x + y + z = 0. ( x R, y R, z R, x + y + z = 0) x R, ( y R, z R, x + y + z = 0) x R, y R, ( z R, x + y + z = 0) x R, y R, z R, (x + y + z = 0) x R, y R, z R, x + y + z 0
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Negate the statement “Every integer can be written as the sum of three squares.” ( n Z, r, s, t Z, n = r 2 + s 2 + t 2 ). n Z, ( r, s, t Z, n = r 2 + s 2 + t 2 ). n Z, r, s, t Z, (n = r 2 + s 2 + t 2 ). n Z, r, s, t Z, n r 2 + s 2 + t 2. Is the original statement true?
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Universal Conditional Statements A universal conditional statement is of the form x S, P(x) Q(x). The converse is x S, Q(x) P(x). The inverse is x S, P(x) Q(x). The contrapositive is x S, Q(x) P(x).
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Negation of Universal Conditional Statements Negate the statement x R, x < 10 x 2 < 100. ( x R, x < 10 x 2 < 100) x R, (x < 10 x 2 < 100) x R, (x < 10) (x 2 100). Which one is true?
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Putnam Question A-2 (1981) Two distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of the squares of C with all the integers 1, 2, …, 64, there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap g.
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Putnam Question A-2 (1981) Consider the standard numbering 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 Note that the largest difference is 9.
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Putnam Question A-2 (1981) Could the answer be 9? 9 is the largest C-gap if 9 is a C-gap, and 10 is not a C-gap.
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Putnam Question A-2 (1981) 10 is not a C-gap if There exists a numbering of the squares such that no two adjacent squares differ by at least 10. Equivalently, there exists a numbering of the squares such that every two adjacent squares differ by at most 9. We have just seen that this is true. Therefore, 10 is not a C-gap.
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Putnam Question A-2 (1981) Is 9 a C-gap? Consider the two squares that are labeled #1 and #64. There is a path of at most 8 squares linking square #1 and square #64. Of the 7 differences along this path, one must be at least 9, since the total difference is 63. Therefore, 9 is a C-gap.
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