: 91-90 (90/12/21) : 91/1/20 : 1 !" : ( [SW:Assignment#1] / ' 0 . [email protected] - . 91/1/20 . '. * '( 1 2 3 '24 # . '. * '( 1 2 3 '24 * < ) *# $ . 9$ : '" ) ; : '" . . = * '( ; < > ;? 1 . '= H # $ * ) I JC G3 F G D@ E:@ A . # $ B ( '. 7 )) : 0 (NF ) . .(( '. (- 7 )) : 0 ( ) .(( '. YG 1 = . =.[ " 3 O G '< 7 )) : X % & '( ) *# $ $ (5 . 2 . ) *7 2 8& 2 ) ( :@ ' ) . C D 1 ! : '" ( ) E(M) E( ) E(NF ) ! IO .1 Q. 3 0 (M) .(( 'R (- 7 )) : 3 O .G TUV W ' ! "# G 0 ( ).(( (- *0 7 D S; * / IC < . = . " x + y = x − y" U2C 1 * / 4 Q(x, y) 0 y Ex Z ; G G =) ' P ) * 2U ] I ) 8 F H .) '. * '( : '" O . D ) * ' ; D S ; .2 * \D .((G3 a) Q(1,1) b) ∀y Q(1, y) c) ∃x Q(x, 2) d) ∃x∃yQ(x, y) e) f) g) h) ∀x∃yQ(x, y) ∃y∀xQ(x, y) ∀y∃xQ(x, y) ∀x∀yQ(x, y) . . .G 1 .G 1 9< = N = 1 g4 ) *1 W4 D 'e / . cd= ' e 'U0 D EN = 1 g4 W ) *h _` D 1 ! 'C CE971 ] 8f & ECE971 'e 2* R 1 W4 8f & a * a a E 'e 'C CE971 . D :a2C .3 'b4 (a 'b4 * (b )'b4 * (c a 'b4 .G = (d $ %& o o o o : 91-90 (90/12/21) : 91/1/20 : 2 !" EN = 1 g4 ) *1 W4 2* ) *h D W 8f & EG 1 $1 ? 'b4 .G . * (e = g. > D ABOX .4 A ={A(d), A(e), A(f), B(f), r(d, e), r(e, g),s(e, f ), s(g, g), s(g, d), r(g, d), s(g, f )} . a) b) c) d) e) f) G3 F 3* 'I! /- D ) '2 A∐B ∃s. ¬A ∀s.A ∃s. ∃s. ∃s. ∃s.A ¬∃r.(¬A ∏ ¬B) ∃s.(A ∏ ∀s.¬B) ∏ ¬∀r.∃r.(A ∐ ¬A) . 3* R ) R * * ! 2* D ALC D '" *! *) . * / 4 E'U. \ D 1 ! . .5 a) ∀r. ∀s.A ∏ ∃r. ∀s.B ∏ ∀r. ∃s.C ⊑ ∃r. ∃s.(A ∏ B ∏ C) b) ∀r. ∀s.A ∏ (∃r. ∀s. ¬A ∐ ∀r. ∃s.B) ⊑ ∀r. ∃s.(A ∏ B) ∐ ∃r. ∀s. ¬B . g. > D : '" . I = ( ∆, I) R J. *! D * R .6 ∆ = {t1 , t 2 , f1 , f 2 , c1 , c2 , j, k , l, m , n} Person I = {j, k, l, m, n} Car I = {t1 , t 2 , f1 , f 2 , c1 , c2 } Ferrari I = {f1 , f 2 } Toyota I = {t1 , t 2 }| likes I = {( j, f1 ), (k, f1 ), (k, t 2 ), (l, c1 ), (l, c 2 ), (m, c1 ), (m, t 2 ), (n, f 2 ), (n, c2 )} .( a) b) c) d) H 8& 2*) $I D ∃likes.Ferrari ∏ ∃likes.Toyota ∃likes.Ferrari ∏ ∀likes.Ferrari ∃likes.Ferrari ∏ ∃likes. ¬Ferrari ∃likes.Cars ∏ ∀likes.¬(Toyota ∐ Ferrari) . g. > D ABOX .7 : 91-90 (90/12/21) : 91/1/20 : 3 !" HasChild( joe, ann) HasChild( joe, eva) HasChild( joe, mary) (≤ 2HasChild)( joe) S ; . ABOX 7 -. / . ? / ) R EG GR% .( دھ 'C < YG 9$ ? ABOX 7 )؟G 9$ ? m (UNA) W + (,- !. : -G . NP 1 g D ) *1 m< M G( (A → B) ↔ ¬(A ∧ ¬B) -NF ((A → B) → (A → C)) → ((A → (B → C)) - .8 :G 'g 2* 1 m< 7 E'U. 'gF D 1 ! . .9 (¬A ∨ B) → (B → A) -NF A → ((B → C) → ((A → B) → (A → B) → (A → C)) * / 4 E'U. 'gF D 1 ! . .10 {P ∨ Q, P → (S ∧ R), Q → (S ∧ R)} ⊢ ¬(¬S ∨ ¬R) g pe / . . 7 ( $ E !" ! "' _` U" ) * <o . . =[ 4 8 & 2m ) r2 [ 4 E 1 = r q .(G3 : rmC . 7 ( $ . )D : '" . ! "' _` 1 ' ( 7 .11 N "' ) . _` U" ) *1 ' ( G ! _` D 1 ' ( * U ) * <o
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