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# as2 - y in A there exists a z in A such that x y =...

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Predicate Logic Practice 1. Let P ( X ) and Q ( X ) represent “ X is a rational number” and “ X is a real number” respectively. Symbolize the following sentences: a. Every rational number is a real number. b. Some real numbers are rational numbers. c. Not every real number is a rational number. 2. Assume that St. Francis is loved by everyone who loves someone. Also assume that noone loves nobody. Deduce that St. Francis is loved by everyone. 3. An Abelian group is a set A with a binary operator + that has cer- tain properties. Let P ( x,y,z ) and E ( x,y ) represent x + y = z and x = y , respectively. Express the following axioms for Abelian groups symbolically. For every x and
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Unformatted text preview: y in A , there exists a z in A such that x + y = z (closure) • If x + y = z and x + y = w then z = w (uniqueness) • ( x + y ) + z = x + ( y + z ) 4. Consider the following interpretation: Domain = { 1 , 2 } Assignment of constants: a = 1 and b = 2 Assignment of functions: f (1) = 2 and f (2) = 1 Assignment for predicate P : P (1 , 1) = T ; P (1 , 2) = T ; P (2 , 1) = F ; P (2 , 2) = F Evaluate the truth value of following formulas in the above interpre-tation: • P ( a,f ( a )) ∧ P ( b,f ( b )) • ( ∀ x )( ∃ y ) P ( y,x ) • ( ∀ x )( ∀ y )( P ( x,y ) → P ( f ( x ) ,f ( y )))...
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