Give a proof to show:
(∀x)x=f(x,y),(∀x)φ(x,x)⊢(∀x)(x=f(x,y)∧φ(x,x))
Make a proof that demonstrates (∀x)x=f(x,y),(∀x)φ(x,x)⊢(∀x)(x=f(x,y)∧φ(x,x)) where f is a binary function symbol and φ is a binary predicate symbol.
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a...
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
Give a proof to show that for any wffs A, B: (3x)(A A (V)B) F (x)B.
Section 15.1 Worksheet Find the gradient field F = νφ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. φ(x, y)-yx2+ y2 for x2 + y2 2. 9, (x, y) # (0,0)
Section 15.1 Worksheet Find the gradient field F = νφ for the potential function φ. Sketch a few level curves of φ and a few vectors of F. φ(x, y)-yx2+ y2 for x2 + y2 2. 9, (x, y)...
Give a proof to show that for any wffs A,B: (∃x)A→(∀x)B⊢(∀x)(A→B)
Show that the given map is surjective. Please give a detailed,
thorough formal explanation/proof. It's somewhat obvious it is
surjective, but I don't know how to start the proof. We are
supposed to take y element of codomain and show that there exists
f(x) = y but where is the codomain and where is the domain?
Somewhat confused since we have two binary structures. Thanks!
7. (R, :) with (R, :) where 0(x) = x3 for x ER
23. (a) Show that a function f : X → Y is a surjection if and only if there is a funct io On g : Y → X such that fog = idy. (b) Show that a function : X → Y with nonempty domain X is an injection if and only if there is a function g : Y → X such that g o f-idx. How does this result break down if X = φ? (c) Show...
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...
q2 please
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...