Mathematical Foundation of Computer Science

DHARM 124 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE That is, X 1 X 2 .............. ............. ............ k. Xk k+1. A ∴ ...

DHARM PROPOSITIONAL LOGIC 125 Since we find the conclusion; therefore conclusion is valid at stage 3. Thus, conclusion is valid ...

DHARM 126 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE We obtain a contradiction therefore, our assumption is wrong at stage 4. H ...

DHARM PROPOSITIONAL LOGIC 127 Sol. Assume, S : Stephen loves Joyce M : Matrye is happy P : parents are happy L : Shalezi will be ...

DHARM 128 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE Thus we have, /∴ ((A → B) ∧ (B → C)) → (A → C) ((A → B) ∧ (B → C)) /∴ (A ...

DHARM PROPOSITIONAL LOGIC 129 α-formula ß-formula ~ ~ X : α (X ∨ Y) : β X : α 1 β 1 : X | Y : β 2 (X ∧ Y) : α ~ (X ∧ Y) : β X : ...

DHARM 130 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE a b X X X a a b T a 1 T 1 a 2 T 1 b 1 b 2 Fig. 5.27 Now we shall define fe ...

DHARM PROPOSITIONAL LOGIC 131 For example, consider the formula X: (((P → Q) ∧ P) → Q) Then tableaux of X will be, (((P→∧→Q) P) ...

DHARM 132 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE These possibilities are shown below in Fig. 5.29. Or, α θ θ β α 1 α (^2) T ...

DHARM PROPOSITIONAL LOGIC 133 Now construct the tableaux for this formula ∼→∧→(((A B) A) B) A ~A B Moving according to this path ...

DHARM 134 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ~ (((((X 12 ∧∧∧∧→ ∼X ) X ) 3 X )4 5X ) T) T ∼ ∼H ∼T H ∼J ∼H ∼M J ∼∼T M T ∼ ...

DHARM PROPOSITIONAL LOGIC 135 That is, X 1 : True X 2 : True X 3 : True X 4 : True X 5 : True /∴ ~ T : False Therefore argument ...

DHARM 136 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE 5.7.1 Symbolization of Statements using Predicate......................... ...

DHARM PROPOSITIONAL LOGIC 137 In order to determine the truth value of (one-place predicate) statement E(x) : where x is a even ...

DHARM 138 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE This expression is also called a predicate expression or predicate formula ...

DHARM PROPOSITIONAL LOGIC 139 N(y) : ,, “y is a number” Then we translate the statement straightforward by, (∀x) [N(x) → ((∃y) N ...

DHARM 140 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE · Similarly, the truth value of the expression (∀x) (∃y) [E(y) ∧ G(y, x)] ...

DHARM PROPOSITIONAL LOGIC 141 ● (∀∧→xx y z) ((P( ) Q( )) R( ) xis bounded ●(∀∧yxy y) (G( , ) Q( )) xis free yis bounded zis free ...

DHARM 142 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE e.g. l (∀x) P(x) ∨ (∃x) ~ P (x); is a valid predicate formula (VPF) and so ...

DHARM PROPOSITIONAL LOGIC 143 This rule is called universal instantiation and is denoted by UI in the inference theory. Therefor ...