Introduction to Predicates and Quantified Statements II

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Introduction to Predicates and Quantified
Statements II

• Lecture 10
• Section 2.2
• Fri, Feb 2, 2007

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Negation of a Universal Statement

• What would it take to make the statement
  Everybody likes me false?

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Negation of a Universal Statement

• What would it take to make the statement
  Somebody likes me false?

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Negations of Universal Statements

• The negation of the statement
• ?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 the statement
• ?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

• Are these statements equivalent?
• Any investment plan is not right for all
  investors.
• There is no investment plan that is right for
  all investors.

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The Word Any

• We should avoid using the word any when writing
  quantified statements.
• The meaning of any is ambiguous.
• You cant put any person in that position and
  expect him to perform well.

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Negation of a Universal Conditional Statement

• How would you show that the statement
• You cant get a good job without a good
  edikashun
• is false?

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Negation of a Universal Conditional Statement

• The negation of ?x ? S, P(x) ? Q(x) is the
  statement
• ?x ? S, ?(P(x) ? Q(x))
• which is equivalent to the statement
• ?x ? S, P(x) ? ?Q(x).

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Negations and DeMorgans Laws

• Let the domain be D x1, x2, , xn.
• The statement ?x ? D, P(x) is equivalent to
• P(x1) ? P(x2) ? ? P(xn).
• Its negation is
• ?P(x1) ? ?P(x2) ? ? ?P(xn),
• which is equivalent to
• ?x ? D, ?P(x).

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Negations and DeMorgans Laws

• The statement ?x ? D, P(x) is equivalent to
• P(x1) ? P(x2) ? ? P(xn).
• Its negation is
• ?P(x1) ? ?P(x2) ? ? ?P(xn),
• which is equivalent to
• ?x ? D, ?P(x).

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Evidence Supporting Universal Statements

• Consider the statement
• All crows are black.
• Let C(x) be the predicate x is a crow.
• Let B(x) be the predicate x is black.
• The statement can be written formally as
• ?x, C(x) ? B(x)
• or
• C(x) ? B(x).

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Supporting Universal Statements

• Question What would constitute statistical
  evidence in support of this statement?

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Supporting Universal Statements

• The statement is logically equivalent to
• ?x, B(x) ? C(x)
• or
• B(x) ? C(x).
• Question What would constitute statistical
  evidence in support of this statement?

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Algebra Puzzler

• Find the error(s) in the following solution.