Give a proof to show that for any wffs A,B: (∃x)A→(∀x)B⊢(∀x)(A→B)
Need help please 2. Find a model for each of the following wffs. a. 3x (p(x) — 9(x))^VX - p(x) b. 3x Vy p(x, y) ^ 3x Vy - p(x, y)
Give a proof to show: (∀x)x=f(x,y),(∀x)φ(x,x)⊢(∀x)(x=f(x,y)∧φ(x,x))
3. Let f be a continuous function on [a, b] with f(a)0< f(b). (a) The proof of Theorem 7-1 showed that there is a smallest x in [a, bl with f(x)0. If there is more than one x in [a, b] with f(x)0, is there necessarily a second smallest? Show that there is a largest x in [a, b] with f(x) -0. (Try to give an easy proof by considering a new function closely related to f.) b) The proof...
1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why. 2- What is the analogy between Z and F[x]? 3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x]. (a) Perform the following computation in Z5[x]/(p(x)). Give your answers in the form [r(x)] where r(x) has degree as small as possible. i. [4x] + [3x^2 + x + 2] ii. [x^2][2x^2+1] (b) Show...
give a proof by contradiction. there does not exist any rational number x such that x * sqrt(2) = sqrt(3)
Write down how the expression (A-B) V - (CA D)A (E-F) of wffs (an example is given below): Example: The expression (A A-B) V (CD) 1. A is a wff (propositional variable) 2. B is a wff (propositional variable 3. -B is a wff (obtained by applying to 2) 4. AA-B is a wff (obtained by applying A to 1 and 3) 5. (AA-B) is a wff (obtained by applying parentheses to 4) 6. C is a wff (propositional variable)...
Proof or give a counterexample for the following statement Let f:[8,18]. Then f(f-1(G))=G for any G⊆ Justify your answer → * We were unable to transcribe this image
For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Make sure you draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line. (a) (x) (y) (z) ((Fxy --> Gxy) v (Hxa --> Hzbu)) (b) (z) (Ex) (y) (Axayzw v Bxycuvz)
(a) Let f(x) = 3x – 2. Show that f'(x) = 3 using the definition of the derivative as a limit (Definition 21.1.2). 1 (b) Let g(x) = ? . Show that y that -1 g'(x) = (x - 2)2 using the definition of the derivative as a limit (Definition 21.1.2).