 1.1.1: If A and B are sets, show that A ~ B if and only if An B = A.
 1.1.2: Prove the second De Morgan Law [Theorem 1.1.4(b)].
 1.1.3: Prove the Distributive Laws: (a) A n (B U C) = (A n B) U (A n C), (...
 1.1.4: Thesymmetric differenceof twosetsA and B is the setD of all element...
 1.1.5: For each n EN, let An = {(n + l)k : kEN}. (a) What is Al n A2? (b) ...
 1.1.6: Draw diagrams in the plane of the Cartesian products A x B for the ...
 1.1.7: Let A := B := {x EIR: 1 :::x ::: I} and considerthe subset C := {(...
 1.1.8: Let I(x) := l/x2, x ;6 0, x E R (a) Determine the direct image I(E)...
 1.1.9: Let g(x) := x2 and I(x) := x + 2 for x E R and let h be the composi...
 1.1.10: Let I(x) := x2 for x E IR,andlet E := {x E IR: 1 :::x :::O}and F :...
 1.1.11: Let I and E, F be as in Exercise10.Find the sets E\ F and I (E) \1 ...
 1.1.12: Show that if I : A + B and E, F are subsets of A, then I(E U F) = ...
 1.1.13: Show that if I: A + Band G, H are subsets of B, then II(G U H) = ...
 1.1.14: Show that the function I definedby I(x) := x/Jx2 + 1,x E IR, is a b...
 1.1.15: For a, b E IRwitha < b, findan explicitbijectionof A := {x : a < x ...
 1.1.16: Give an example of two functions I, g on IRto IRsuch that I :f:g, b...
 1.1.17: (a) Show that if I: A + B is injective and E ~ A, then 11(f(E = E...
 1.1.18: (a) Suppose that I is an injection. Show that II 0 I(x) =x for all...
 1.1.19: Provethat if / : A ~ B is bijective and g.: B ~ C is bijective,then...
 1.1.20: Let / : A ~ B and g : B ~ C be functions. (a) Show that if go/ is i...
 1.1.21: Prove Theorem 1.1.1
 1.1.22: Let /, g be functions such that (g 0 f)(x) =x for all x e D(f) and ...
Solutions for Chapter 1.1: Sets and Functions
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 1.1: Sets and Functions
Get Full SolutionsIntroduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. Chapter 1.1: Sets and Functions includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Since 22 problems in chapter 1.1: Sets and Functions have been answered, more than 8667 students have viewed full stepbystep solutions from this chapter.

Derivative of ƒ
The function defined by ƒ'(x) = limh:0ƒ(x + h)  ƒ(x)h for all of x where the limit exists

Descriptive statistics
The gathering and processing of numerical information

Dihedral angle
An angle formed by two intersecting planes,

Elements of a matrix
See Matrix element.

Exponential form
An equation written with exponents instead of logarithms.

Graph of a function ƒ
The set of all points in the coordinate plane corresponding to the pairs (x, ƒ(x)) for x in the domain of ƒ.

Graph of a relation
The set of all points in the coordinate plane corresponding to the ordered pairs of the relation.

Graph of an inequality in x and y
The set of all points in the coordinate plane corresponding to the solutions x, y of the inequality.

Identity
An equation that is always true throughout its domain.

LRAM
A Riemann sum approximation of the area under a curve ƒ(x) from x = a to x = b using x1 as the lefthand endpoint of each subinterval

Midpoint (on a number line)
For the line segment with endpoints a and b, a + b2

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Pseudorandom numbers
Computergenerated numbers that can be used to approximate true randomness in scientific studies. Since they depend on iterative computer algorithms, they are not truly random

Quotient rule of logarithms
logb a R S b = logb R  logb S, R > 0, S > 0

Rational expression
An expression that can be written as a ratio of two polynomials.

Reflection
Two points that are symmetric with respect to a lineor a point.

Remainder theorem
If a polynomial f(x) is divided by x  c , the remainder is ƒ(c)

Vertices of an ellipse
The points where the ellipse intersects its focal axis.

xyplane
The points x, y, 0 in Cartesian space.

Ymin
The yvalue of the bottom of the viewing window.