PROLOG Exercise 7 Consider the following program. a(X):- b(X), c(X), d(X). a(X):- c(X), d(X). a(X):- d(X). b(1). b(a). b(2). b(3). d(10). d(11). c(3). c(4). Given the query ?- a(X). Write out the variable bindings that the variable X would get when the above query is run. Exercise 4 Indicate whether the following are syntactically correct Rules. a :- b, c, d:- e f. % Is this syntactically correct? happy(X):- a , b. % Is this syntactically correct? happy(X):- hasmoney(X) &...
A subset D of a metric space (X, d) is dense if every member of X is a limit of a sequence of elements from D. Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense subset of X. 1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d, and V(z) 0 elsewhere. We will use the variational wave function v(z) = A(b + r), t(x)-A(b-x), -b < r < 0, 0 < x < b, to show that a bound state exists for any U0. a) Normalize the wave function. Find the expectation values of the kinetic and potential energies b) Show that for sufficiently large b, with b> d, the expectation...
Let (X, d) be a metric space, and let ACX be a subset (a) (3 pts) Let x E X. Write the definition of d(x, A) (b) (7 pts) Assume A is closed. Prove that d(x,A-0 if and only if x E A. Let (X, d) be a metric space, and let ACX be a subset (a) (3 pts) Let x E X. Write the definition of d(x, A) (b) (7 pts) Assume A is closed. Prove that d(x,A-0 if...
In the following, (X,d) is an arbitrary metric space and (X,d,μ) is an arbitrary metric measure space. (6) Recall the definition of bounded set: The set A C (X, d) is bounded if δ(A) < 00 where 6(A)p d(a,a). (X,d) with ACBand B is bounded then A is bounded (a) Show that if A, B (b) Fix a set A. I B - (r), a single point, show that D(A, B)-0 if and only f (c) Prove that the function...
Let X = {a,b,c,d,e) and T = {X, Ø, {a}, {c,d}, {a,c,d}, {b,c,d,e }} and {A= {b,c,d {interior(A)= {cd a {interior(A)= {a,c,eb {interior(A)= {d.c {interior(A)= {a,c,d .d
Evaluate the double integral ∫∫D x cos y dA, where D is bounded by x = 0, y = x², and x = 3 Answer:
2. [2+9+6=17] Let X be a nonempty set. Two metrics d and d on X are said to be uniformly equivalent if the identity map from (X, d) to (X, d) a nd its in- verse are uniformly continuous. (a) Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X. (b) Let (X,d) and (x, ) be uniformly equivalent. Are the following true or false? (i) If (X, d) is bounded, then must also...
R(Xwhere the degree of R(x) is less than the degree of D(x) D(x) Use polynomial division to rewrite each expression in the form Q(x) X 8 (a) X 12 16x 9 (b) _ 2x 1 12x2 (c) x2 12 4x 1
4. (Suggested by Keenan) Let d: X X X + Rxo be a semimetric on a set X. We say that d is an ultrametric if it satisfies the strong triangle inequality: d(21, 12) <max{d(21,13), (13,12) }; for all 11, 12, 13 e X. (a) hamuga prottu mupretatus காபாடியமாடு படியy: (b) Let (x, d) be an ultrametric space. Show that every triangle in (x,d) is isosceles, that is, if d(11,13) <d(22, 23) and d(11,12)<max{d(11, 13), d(13, 22), then d(21, 23)...