(x'). (15 pts) to prove or disprove sqrt(2 + sqrt(6x)) = the definition of
(-8,00) defined as f(x)= x + 6x +1. Prove or disprove that it is 1-1 11. We are given a function S:(-3,00) and/or onto, using algebraic evidence.
Prove or Disprove #3
(d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a)
(d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a)
Question 8 f(x) = 3x2 + 2 Find f'(x). 0 sqrt(6x) O (3x^2 + 2)^3/2 O 1/sqrt(3x^2 + 2) O 3x/sqrt(3x^2 + 2)
Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n)), and g(n) = O(n2), then f(n) + g(n) = O(n5). Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n), and g(n) = O(n2), then fin)/g(n) = O(n).
Prove or disprove by using Definition 2.1.3
for any n E N. Then {ann is a convergent (g) Let an = sequence. (h) Let an sequence." for any n E N. Then {an} is a convergent
Prove or disprove that U(15) and U(20) are isomorphic.
(1) Prove or disprove that if all the elements of a matrix A is
even, the determinant of A is even.
(2) Compute the following determinant
(1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c
Prove or Disprove 23. If x <y, then-y -x. Remark 39. What is the analogous problem and answer for bounded below and infima? Prove or Disprove 37. Let SCR be nonempty and bounded below. Then inf(S) exists.
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
(3 + 3 = 6 pts.) Prove or disprove the following statements. If you are proving a statement, then give proper reasoning. If you are disproving a statement, then it is enough to give an example which demonstrates that the statement is false. i. If A and B are two n x n matrices, then (A + B)2 = A + 2AB + B2. ii. Let A be a nxn matrix and let I be the n x n identity...