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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 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) y, x, P(x, y) y, x, P(x, y) x, y, P(x, y) Legend: A B is valid
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))
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.
Translations • y x Q(x, y). • x1 x2 ( (x1 x2) Q(x1 , Jeopardy) Q(x2 , Jeopardy) )
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)).
Characters • ≥ ≡ • • • • •
