Tables. Part I. Truth-Tables P Q P Q P Q P P Q P Q

Transcription

1 Tables art I Truth-Tables Q Q Q Q Q

2 372 TABLES FOR ART I Equiences for connectives Commutativity: Associativity: Q == Q, ( Q) R == (Q R), Q == Q, ( Q) R == (Q R), Q == Q ( Q) R == (Q R) Idempotence: Double Negation: ==, == == Inversion: True/False-elimination: True == False, True ==, False == True False == False, True == True, False == Negation: Contradiction: == False == False Excluded Middle: == True Distributivity: De Morgan: (Q R) == ( Q) ( R), ( Q) == Q, (Q R) == ( Q) ( R) ( Q) == Q Implication: Contraposition: Q == Q Q == Q Bi-implication: Self-equience: Q == ( Q) (Q ) == True

3 TABLES FOR ART I 373 Weakening rules - -weakening: Q ==, Extremes: == Q False ==, Monotonicity: == True If == Q, then R == Q R, If == Q, then R == Q R Bound Variable: Equiences for quantifiers x [ : Q] == y [ [y for x] :Q[y for x]], x [ : Q] == y [ [y for x] :Q[y for x]] Domain Splitting: x [ Q : R] == x [ : R] x [Q : R], x [ Q : R] == x [ : R] x [Q : R] One-element: Empty Domain: x [x = n : Q] == Q[n for x], x [False : Q] == True, x [x = n : Q] == Q[n for x] x [False : Q] == False Domain Weakening: De Morgan: x [ Q : R] == x [ : Q R], x [ : Q] == x [ : Q], x [ Q : R] == x [ : Q R] x [ : Q] == x [ : Q]

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5 TABLES FOR ART II 375 art II Derivation rules for and -introduction: -elimination: Q { -intro on and: } Q Q { -elim on : } resp Q -introduction: -elimination: (m 1) { Assume: } Q { -intro on and(m 1): } Q Q { -elim on Q and: }

6 376 TABLES FOR ART II Derivation rules for, False and -introduction: -elimination: (m 1) { Assume: } False { -intro on and(m 1): } { -elim on and: } False False-introduction: { False-intro on and: } False False-elimination: False { False-elim on : } -introduction: -elimination: { -intro on : } { -elim on : } Note: -elim and False-intro are similar, but differ in use See Section 143, I

7 TABLES FOR ART II 377 Derivation rules for and -introduction: (m 1) (m 1) { Assume: } Q { -intro on and(m 1): } Q ----resp---- Q { Assume: } { -intro on and(m 1): } Q -elimination: Q { -elim on and: } Q ----resp---- Q Q { -elim on and: } -introduction: Q Q { -intro on and: } Q -elimination: Q { -elim on : } Q resp Q

8 378 TABLES FOR ART II roof by contradiction { Assume: } (m 1) False { -intro on and(m 1), followed by -elim: } roof by case distinction (n) Q R Q R { Case distinction on, and: } R

9 TABLES FOR ART II 379 Derivation rules for and -introduction: -elimination: (m 1) { Assume: } var x; (x) Q(x) { -intro on and(m 1): } x [ (x) :Q(x)] x [ (x) :Q(x)] (a) { -elim on and: } Q(a) (a must be an object being available in line ) -introduction: -elimination: { Assume: } (m 1) False x [ (x) : Q(x)] { -intro on and(m 1): } x [ (x) :Q(x)] x [ (x) :Q(x)] x [ (x) : Q(x)] { -intro on and: } False

10 380 TABLES FOR ART II Alternative derivation rules for -introduction: -elimination: (a) Q(a) { -intro on and: } x [ (x) :Q(x)] x [ (x) :Q(x)] { -elim on : } ick an x with (x) andq(x) (x in line mustbe new ) (a must be an object being available in lines and)

11 TABLES FOR ART III 381 art III Sets A B def = x [x A : x B] A = B def = A B B A A B def = A B def = {x U x A x B} {x U x A x B} A C def = {x U (x A)} A\B def = {x U x A (x B)} U = { x U True} = { x U False} roperty of : t {x D (x)} == t D (t) roperty of : roperties of = : A B t A == t B A = B == x [x A x B] A = B t A == t B A = B t B == t A roperty of : roperty of : t A B == t A t B t A B == t A t B roperty of C : roperty of \ : t A C == (t A) t A\B == t A (t B) roperties of U : roperties of : t U == True t == False A = U == x [x A : True] A = == x [x A : False] roperty of : roperties of : C (A) == C A (a, b) A B == a A b B (a, b) =(a,b ) == a = a b = b

12 382 TABLES FOR ART III Mappings roperty of mapping F : A B : x [x A : 1 y[y B : F (x) =y]] Image and source Let F : A B be a mapping, A A and B B the image of A : the source of B : F (A )= F (B )= {b B x [x A : F (x) =b]} {a A F (a) B } roperties of image : roperty of source : x A == F (x) F (A ) x F (B ) == F (x) B == y F (A ) x [x A : F (x) =y] Special mappings roperty of surjection roperty of injection for F : A B : for F : A B : y [y B : x1,x 2 [x 1,x 2 A : x [x A : F (x) =y]] (F (x 1 )=F(x 2 )) (x 1 = x 2 )] roperty of bijection for F : A B : y [y B : 1 x[x A : F (x) =y]] roperty of inverse function F 1 : B A for bijection F : A B : F (x) =y == F 1 (y) =x roperty of composite mapping G F : A C for F : A B and G : B C : G F (x) =z == G(F (x)) = z

13 TABLES FOR ART III 383 Standard derivation rules for induction Induction: Induction from a Z : A(0) A(a) i [i N : A(i) A(i+1)] { Induction on and: } n [n N : A(n)] i [i Z i a : A(i) A(i+1)] { Induction on and: } n [n Z n a : A(n)] Strong induction: Strong induction from a Z : k [k N : j [j N j<k : A(j)] A] { Strong induction on : } n [n N : A(n)] k [k Z k a : j [j Z a j<k : A(j)] A] { Strong induction on : } n [n Z n a : A(n)]

14 384 TABLES FOR ART III Special relations R on A is reflexive if x [x A : xrx] R on A is irreflexive if x [x A : (xrx)] R on A is symmetric if x,y [x, y A : xry yrx] R on A is antisymmetric if x,y [x, y A :(xry yrx) x = y] R on A is strictly antisymmetric if x,y [x, y A : (xry yrx)] R on A is transitive if x,y,z [x, y, z A :(xry yrz) xrz] R on A is linear if x,y [x, y A : xry yrx x = y] Equience relation R on A is an equience relation if R is reflexive and symmetric and transitive Orderings R on A is a quasi-ordering if R is reflexive and transitive R on A is a reflexive ordering if R is reflexive, antisymmetric and transitive R on A is an irreflexive ordering if R is irreflexive, strictly antisymmetric and transitive R on A is a reflexive linear ordering if R is a reflexive ordering which is also linear R on A is an irreflexive linear ordering if R is an irreflexive ordering which is also linear Let A, R 1 and B,R 2 be irreflexive orderings The corresponding lexicographic ordering A B,R 3 is defined by: (x, y)r 3 (x,y )ifxr 1 x (x = x yr 2 y )

15 TABLES FOR ART III 385 Special relations and orderings: overview A is a set, R is a relation on A equi- quasi- (partial) linear ence ordering ordering ordering R on A is: relation refl irrefl refl irrefl reflexive irreflexive symmetric antisymmetric str antisymm transitive linear examples: =, <, > == ==

16 386 TABLES FOR ART III Extreme elements Let A, R be a reflexive ordering and A A m A is a maximal element of A if x [x A : mrx x = m] m A is a minimal element of A if x [x A : xrm x = m] m A is the maximum of A if x [x A : xrm] m A is the minimum of A if x [x A : mrx] Let A, S be an irreflexive ordering and A A m A is a maximal element of A if x [x A : (msx)] m A is a minimal element of A if x [x A : (xsm)] m A is the maximum of A if x [x A x m : xsm] m A is the minimum of A if x [x A x m : msx] Upper and lower bounds Let A, R be a reflexive ordering and A A b A is an upper bound of A if x [x A : xrb] a A is a lower bound of A if x [x A : arx]