Mathematical and Logical Foundations of Computer Science Second Class Test Question 1 [Linear Algebra] (a) Consider the following two vectors in R3: ~v = 12 −1 ~w = 31 0 (i) Show that ~v and ~w are linearly independent of each other. [2 marks] (ii) Find a third vector ~u so that {~v, ~w, ~u} form a basis of R3. [2 marks] (b) The points P1 = 31 −3 P2 = 10 −2 P3 = 20 0 are the corners of a triangle in R3. (i) Show that the triangle has a right angle, and say at which corner it occurs. [4 marks] (ii) The triangle defines a plane E in R3. Give its parametric representation and its normal form. [4 marks] (iii) A line L in R3 is given by X = 02 1 + s · 03 0 . Compute its point of intersection with the plane E from the previous item. [4 marks] (c) Let B = {~v1, ~v2, . . . , ~vn} be a basis for an algebra of vectors V , and let ~w be an arbitrary vector in V . (i) When do we say that scalars a1, a2, . . . , an are the coordinates of ~w with respect to B? [1 mark] (ii) Prove that the coordinates of ~w with respect to B are uniquely determined. [3 marks] Question 2 [SAT & Predicate Logic] (a) (i) Let p0, p1, q0, q1, r0, r1 be atoms capturing the states of three cells called p, q, and r , that can each either hold a 0 or a 1: pi captures the fact that cell p 2 35324 LC Mathematical and Logical Foundations of Computer Science holds the value i , and similarly for the other atoms. Consider the following formula: (p0 ∨ p1) ∧ (q0 ∨ q1) ∧ (r0 ∨ r1) ∧ (¬p0 ∨ ¬p1) ∧ (¬q0 ∨ ¬q1) ∧ (¬r0 ∨ ¬r1) ∧(p0 ∨ q0 ∨ r0) ∧ (p1 ∨ q1) ∧ (p1 ∨ r1) ∧ (q1 ∨ r1) Using DPLL, prove whether the above formula is satisfiable or not. Detail your answer. What property of the three cells p, q, and r , is this formula capturing? [4 marks] (ii) Given a CNF (Conjunctive Normal Form) that contains a clause composed of a single literal, can it be proved using Natural Deduction? Justify your answer. [2 marks] (b) Consider the following domain and signature: Domain: N Function symbols: zero (arity 0); succ (arity 1); ∗ (arity 2) Predicate symbols: even (arity 1); odd (arity 1); = (arity 2) We will use infix notation for the binary symbols ∗ and =. Consider the following formulas that capture properties of the above predicate symbols: let S1 be ∀x.(even(x)→ ∃y .x = 2 ∗ y) let S2 be ∀x.((∃y .x = succ(2 ∗ y))→ odd(x)) let S3 be ∀x.∀y .(x = y → succ(x) = succ(y)) where for simplicity we write 0 for zero, 1 for succ(zero), 2 for succ(succ(zero)), etc. (i) Provide a constructive Sequent Calculus proof of: S1, S2, S3 ` ∀x.(even(x)→ odd(succ(x))) [6 marks] (ii) Provide a model M such that M ∀x.(even(x)→ odd(succ(x))) [2 marks] (iii) Provide a model M such that ¬ M ∀x.(even(x)→ odd(succ(x))) [2 marks] (c) Let p be a predicate symbol of arity 1 and q be a predicate symbol of arity 2. Let F be the Predicate Logic formula (∀x.(p(x) ∧ ∃y .q(x, y)))→ ∀x.∃y .(p(x) ∧ q(x, y)). Provide a constructive Natural Deduction proof of F . You are not allowed to make use of further assumptions so all your hypotheses should be canceled in the final proof tree. [4 marks]