L3[1] - Translating Statements a) A student in your class...

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Translating Statements a) A student in your class has a cat, a dog and a ferret. b) All students in your class have a cat, dog, or ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, there is a student in your class who has one of these animals. Let C(x): “x has a cat”, D(x): “x has a dog”, F(x): “x has a ferret” Domain: All students in your class. **Note: In (d) the first answer reads “All students in the class do not have all three animals.” The second answer reads “There is no student in the class that has all three animals.” But these two statements say the same thing, in different ways, and both are valid. ( ( ) ( ) ( )) x C x D x F x ( ( ) ( ) ( )) x C x D x F x ( ( ) ( ) ( )) x C x F x D x ( ( ( ) ( ) ( ))) or, equivalently ( ( ) ( ) ( )) x C x D x F x x C x D x F x ( ) ( ) ( ) xC x xD x xF x
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Quantifiers with Restricted Domains What do these mean? (the domain is real numbers) a) “For all real numbers x restricted to x<0, it is the case that x 2 >0. b) “For all real numbers y other than 0, it is the case that y 3 is not equal to 0. c) “There exists some real number z restricted to z>0 such that z 2 =2. 2 3 2 ) 0 0 ) 0 0 ) 0 2 a x x b y y c z z 2 3 2 same as 0 0 same as 0 0 same as 0 2 x x x y y y z z z
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Precedence of Quantifiers The quantifiers have higher precedence than the operators we have discussed previously. Example: is the disjunction (“or”) of More specifically, it is to be interpreted as and not , , ! ( ) ( ) xP x Q x ( ) and ( ). xP x Q x ( ( )) ( ) xP x Q x ( ( ) ( )). x P x Q x
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Binding Variables When a quantifier is used on a variable x, we say that this occurrence of x is bound . A variable that is not bound to a quantifier, and does not have any particular value assigned to it is said to be free . The part of a logical expression to which a quantifier is applied is called the scope of that quantifier.
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Example Which variables are bound? free? What is the scope of the quantifier?
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L3[1] - Translating Statements a) A student in your class...

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