Use the rules of deduction in the Predicate Calculus (but avoiding derived rules) to find formal proofs for the following sequents: (a) x) F)~(Vx)~ F(x) (b) (Vz) ~ F(x) B) F() (3x)(G(z) Л (Vy) (F(y)...
Use the formal rules of deduction of the Propositional Calculus to carefully prove the following sequents. Feel free to use earlier sequents in proofs of later ones by applying Sequent Introduction. (iv) Q ⇒ R ⊢ (P ∨ Q) ⇒ (P ∨ R)
1. Provide semi-formal Natural Deduction proofs of the following claims. You may only use the eight Natural Deduction inference rules. (a) (PAQ) + R,PAS,-QER (b) XA (X+(Z AY))-XAY (c) F(X A (X (ZAY)))(X AY) (d) AABEBV(A -C) (e) (KVL) +N, KAMENAM (f) (AAB) →C,B,AA-DECAD (g) (AAB) C,BF (AAD)+(CAD) (h) -P→ (QAR)F(PAS) → (RAS) (i) Z-X,ZAYE-XVY
Find faults in the following arguments, with brief explanations: (a) First faulty argument: 2(2) (3x) F(x) 3 (3) F(a) (4) F(a)→G(a) 1,3 (5) G(a) 1,3 (6) (Vr) G(x) 1,2 (7) (Vr) G(x) 3,4 MP 2,3,6 3 E (b) Second faulty argument: (1) (yz)(33) H(z, y) 1 (2) y) H(a, y) (3) (4) 5(5) 4.5 (6) 4.5 (7) (3) H(by) H(a, b) H(b, a) H (a, b) ЛНФа) (3y)(H(a,y) ΛΗ(y, a)) 4,5 л! 6ヨ1 Now find models to demonstrate that the...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
Use rules #1-8 to provide logical proofs with line-by-line justifications for the following arguments. (Note: you can use any of the rules that you wish; however, it is possible to solve the arguments in this section by using only the first 8 rules.) 1 1. E > (A & C) 2. A > (F & E) 3. E /F 2 1. (L v T) > (B & G) 2. L & (K = R) /L & B 3 1. (X...
Calculus III
1) Identify each of the following surfaces: а) z' %3x? - 5у" b) z 4x2-4 y c) z2+3x2-5y = 4 d) z2x23-5y e) у3х* 2) Find and classify all of the critical points for f(x, y)=xy -x2 + y'. 3) Find the maximum and minimum values of f(x,y)=xy over the ellipse х* + 2y %3D1. 4) Let fx, y) x3 -cosy a) Find the first order Taylor polynomial for f(x,y) based at (1,7). b) Find the sccond order...
Find the derivative of the following functions: Inx 22 C. a. F(x) = In (3x) b. Y=: Y=52x-1 d. Y= log1032 e. Y=In(x2-2)2/3 f. Y= In g. Y= log2 (2x - 1) h. Y= 8** e 1+e+
Use the gradient rules to find the gradient of the given function, f(x,y,z) = x+yz y+xz Choose the correct answer below. 1 O A. Vf(x,y,z) = -((1-z?)z(z2 - 1).y? - x?) (y + xz)? OB. Vf(x,y,z) = (z(1-z?)y(z? - 1),z2 + x2) (x + yz)? O c. Vf(x,y,z) = (y(1+z2),x(z? + 1).y? - z?) (x + yz)? OD. Vf(x,y,z) = -(y (1-2²), x(2² - 1), y² - x²) (y + xz)2
2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y E A then x = y. That is Vz, y A f(x) = f(y) → x = y. Which of the following functions are injective? In each case explain why or why not i. f:Z-Z given by f() 3r +7 (1 mark ii. f which maps a QUT student number to the last name of the student with that student...
1. (20 points) Find derivatives of the following functions. (a) f(x) = 1012 (b) g(x) = (ln(x2 + 3)] (c) h(x) = Vx+V2 (d) y=et +e? – x-e