Define injections g : [0, 1) → [0, 1] and h : [0, 1] → [0, 1). Prove that that the functions g and h that you define are injections. It follows from the Cantor-Schroeder-Bernstein theorem that there exists a bijection f : [0, 1] → [0, 1). (Can you see how to define such a bijection directly?)

Let T be a linear transformation from R3 into R such that T(1,1,1)=1, T(1,1,0)=2 and T(1,0,0)=3. Find T(0,1,1)

Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.

Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].

Let x and y be in Z, not both zero, then x2+y2Z+.

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.

If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]