Lecture 2

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Lecture 2
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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, x2 gt 0.
• ?x ? 1, , 10, x2 gt 100.
• ?x ? R, x3 x ? 0.
• ?x ? R, ?y ? R, x2 xy y2 ? 0.
• ?x ? ?, x2 gt 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 Existential Statements

• ?x ? 1, , 10, x2 gt 0.
• ?x ? 1, , 10, x2 gt 100.
• ?x ? R, x3 x ? 0.
• ?x ? R, ?y ? R, x2 xy y2 ? 0.
• ?x ? ?, x2 gt 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, x2 gt 10 is false, then ?x ? R, x2 ?
  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, x2 lt 0 is false, then ?x ? R, x2 ?
  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

• Multiply quantified universal statements
• ?x ? S, ?y ? T, P(x, y)
• The order does not matter.
• Multiply quantified 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 positive integer can
  be written as the sum of three squares.
• ?(?n ? Z, ?a, b, c ? Z, n a2 b2 c2).
• ?n ? Z, ?(?a, b, c ? Z, n a2 b2 c2).
• ?n ? Z, ?a, b, c ? Z, ?(n a2 b2 c2).
• ?n ? Z, ?a, b, c ? Z, n ? a2 b2 c2.
• 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 lt 10 ? x2 lt 100.
• ?(?x ? R, x lt 10 ? x2 lt 100)
• ? ?x ? R, ?(x lt 10 ? x2 lt 100)
• ? ?x ? R, (x lt 10) ? (x2 ? 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

• 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.

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Putnam Question A-2 (1981)

• One possible numbering and path

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Putnam Question A-2 (1981)

• 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.