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2_-_Solutions - Assignment 2 Solutions 1 Negate the...

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Assignment 2 – Solutions 1. Negate the expression x y, ( P ( x ) OR Q ( y )) and simplify your answer. Solution: NOT x y, ( P ( x ) OR Q ( y )) x y, NOT ( P ( x ) OR Q ( y )) x y, (NOT P ( x )) AND (NOT Q ( y )) 2. For each of the following statements, determine whether it is true when the universe of discourse is Z and whether it is true when the universe of discourse is R . (a) x y ( x = y 3 ) Solution: This is false in Z , for example when x = 2 there is no integer y for which x = y 3 . On the other hand, it is true in R . Indeed given x R we can choose y = 3 x and then we have x = y 3 . (b) x y ( y < x AND z ( z < x = z y ) ) Solution: This is true in Z . Indeed given x Z we can choose y = x - 1. Then we have y < x , and for all z Z , if z < x then we have z x - 1, that is z y . On the other hand, the given statement is false in R . Indeed given x R and given y R with y < x , we can choose z R with y < z < x (for example we can choose z = x + y 2 ).
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