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Nested Quantifiers

Nested Quantifiers. Nested Iteration. Let the domain be {1, 2, …, 10}. Let P(x, y) denote x > y. x, y, P(x, y) means x, (y, P(x, y) ) Is the above statement true?. Multiple Quantifiers. x,  y, P(x, y). y,  x, P(x, y). y, x, P(x, y). x, y, P(x, y). x, y, P(x, y).

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Nested Quantifiers

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  1. Nested Quantifiers

  2. Nested Iteration • Let the domain be {1, 2, …, 10}. • Let P(x, y) denote x > y. • x, y, P(x, y) means x, (y, P(x, y) ) • Is the above statement true?

  3. Multiple Quantifiers x,  y, P(x, y) y,  x, P(x, y) y, x, P(x, y) x, y, P(x, y) x, y, P(x, y) y, x, P(x, y)  y, x, P(x, y)  x, y, P(x, y) Legend: A B is valid

  4. Translate to English • Let the domain be the real numbers. • x, y, (((x ≥ 0)  (y < 0))  (x – y > 0)) • Is there something wrong with x, (((x ≥ 0)  (y, y < 0))  (x – y > 0))

  5. Translate to Locigal Expression • Let Q(x,y) denote “student x has been a contestant on quiz show y” • The domain for x is all students at UCSB. • The domain for y is all quiz shows on TV. • Express as a logical expression • Every TV quiz show has had a student from UCSB as a contestant. • At least 2 students from UCSB have been contestants on Jeopardy.

  6. Translations • y x Q(x, y). • x1 x2 ( (x1  x2)  Q(x1 , Jeopardy)  Q(x2 , Jeopardy) )

  7. Negating Nested Quantifiers Negate x y (P(x, y)  Q(x, y)) so that only predicates are negated. • x y (P(x, y)  Q(x, y)). • x y (P(x, y)  Q(x, y)). • x y  (P(x, y)  Q(x, y)). • x y ( P(x, y)   Q(x, y)).

  8. Characters •    ≥ ≡ •        •    •   •      •        

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