languange( Cont) am e Consider the language with one predicate P(x, y)and function f(x, y). We can view them as oNwith≤andf(x,y)=x·y. Q with <and f(x,y)=x:y o Z with and f(x,y)=x-y
Languange(Cont.) . Example . . Consider the language with one predicate P(x, y) and function f (x, y). We can view them as: 1. N with ≤ and f (x, y) = x · y. 2. Q with < and f (x, y) = x ÷ y. 3. Z with > and f (x, y) = x − y. Yi Li (Fudan University) Discrete Mathematics May 28, 2013 6 / 27
St ructure amp dle Consider this sentence bobby' s father can beat up the father of any other kid on the block
Structure . Example . . Consider this sentence ”Bobby’s father can beat up the father of any other kid on the block”. . Solution . . Let 1. K(x): x is a child on the block and b means Bobby. 2. f (x): x’s father, 3. B(x, y): x can beat up y, 4. Finally, ∀x(K(x) → (¬(x = b) → B(f (b), f (x)))). Yi Li (Fudan University) Discrete Mathematics May 28, 2013 7 / 27
St ructure amp dle Consider this sentence bobby' s father can beat up the father of any other kid on the block Solution Let
Structure . Example . . Consider this sentence ”Bobby’s father can beat up the father of any other kid on the block”. . Solution . . Let 1. K(x): x is a child on the block and b means Bobby. 2. f (x): x’s father, 3. B(x, y): x can beat up y, 4. Finally, ∀x(K(x) → (¬(x = b) → B(f (b), f (x)))). Yi Li (Fudan University) Discrete Mathematics May 28, 2013 7 / 27
St ructure amp dle Consider this sentence bobby' s father can beat up the father of any other kid on the block Solution Let o K(x): x is a child on the block and b means Bobby
Structure . Example . . Consider this sentence ”Bobby’s father can beat up the father of any other kid on the block”. . Solution . . Let 1. K(x): x is a child on the block and b means Bobby. 2. f (x): x’s father, 3. B(x, y): x can beat up y, 4. Finally, ∀x(K(x) → (¬(x = b) → B(f (b), f (x)))). Yi Li (Fudan University) Discrete Mathematics May 28, 2013 7 / 27
St ructure amp dle Consider this sentence bobby' s father can beat up the father of any other kid on the block Solution Let O K(x): x is a child on the block and b means Bobby o f(x):xs father
Structure . Example . . Consider this sentence ”Bobby’s father can beat up the father of any other kid on the block”. . Solution . . Let 1. K(x): x is a child on the block and b means Bobby. 2. f (x): x’s father, 3. B(x, y): x can beat up y, 4. Finally, ∀x(K(x) → (¬(x = b) → B(f (b), f (x)))). Yi Li (Fudan University) Discrete Mathematics May 28, 2013 7 / 27