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# 7 - Quantiﬁers Let P(x be a statement depending on x Then...

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Quantifiers Let P ( x ) be a statement depending on x . Then • ∀ x , P ( x ) is “For all x , P ( x ) is TRUE”. • ∃ x , P ( x ) is “There exists an x such that P ( x ) is TRUE”. Example. Let P ( x ) be “ x 2 = 2”. Is x , P ( x ) TRUE or FALSE? Is x , P ( x ) TRUE or FALSE? 1

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Example. Let P ( x ) be “ x 2 = 2”. A = x , P ( x ). B = x , P ( x ). We need to specify the “ universe of discourse for x . 1. U of D = Q . A and B are FALSE. 2. U of D = { 2 , - 2 } . A and B are TRUE. 3. U of D = R . A is FALSE, B is TRUE. 2
If U of D is P then x, y x ” is equivalent to “ y = 1”. ( y is a least element in P ). y x, y x ” is TRUE. (There is a least element in P ). y x, y x ” is FALSE. (Any positive integer is least in P ). 3

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Let S be some U of D. Then x S , P ( x )” means “For all x from S , P ( x ) is TRUE” x S , P ( x )” means “There is x from S such that P ( x ) is TRUE”” Examples. x Q , x 2 = 2” is FALSE.
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7 - Quantiﬁers Let P(x be a statement depending on x Then...

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