Consider the space with the Euclidean norm ||x||2 = √|x1|2 + |x2|² (i) Give the definitions of an open set and closed set in R2. What is a bounded set in R²? [10 Marks] (ii) Identify which of the following sets in R² are open and which are not. Are any of these sets bounded? (a) A = (2, 4) x (1, 2), (b) B (2, 4) {1,2}, (c) C = (2,4) × R. C.11 [10 Marks] [10 Marks] [10 Marks]

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.

26. Let and. Prove that for any subset of T of .

A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.