Nested Quantifiers

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Nested Quantifiers
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Nested Iteration

• Let the domain be 1, 2, , 10.
• Let P(x, y) denote x gt y.
• ?x, ?y, P(x, y) means ?x, (?y, P(x, y) )
• Is the above statement true?

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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
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Translate to English

• Let the domain be the real numbers.
• ?x, ?y, (((x 0) ? (y lt 0)) ? (x y gt 0))
• Is there something wrong with
• ?x, (((x 0) ? (?y, y lt 0)) ? (x y gt 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.

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Translations

• ?y ?x Q(x, y).
• ?x1 ?x2 ( (x1 ? x2) ?
• Q(x1 , Jeopardy) ?
• Q(x2 , Jeopardy)
• )

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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)).

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Characters

• ? ? ?
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?
• ? ? ? ? ?
• ? ? ? ? ? ? ? ?