Chapter 2.3, Problem 9ES

Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.

Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.

Definition: A topology on a set X is a collection T of subsets of X having the following properties. (1) Both the empty set and X itself are elements of T. (2) The union of an arbitrary collection of elements of T is an element of T. (3) The intersection of a finite number of elements of T is an element of T. A set X with a specified topology T is called a topological space. The subsets of X that are members of are called the open sets of the topological space.

1) If f(x) = g¹ (g(x) + a) for some real number a and invertible function g, show that f(x) = (fo fo... 0 f)(x) = g¯¹ (g(x) +na) n times for all integers n ≥ 1.

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u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

Question 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.

39 Two sides of one triangle are congruent to two sides of a second triangle, and the included angles are supplementary. The area of one triangle is 41. Can the area of the second triangle be found?