Schaum's Outline Series included a collection of over examples and solved problems, each specifically designed to . Theory Tables and Matrices. Schaum's outline of theory and problems of set theory and related topics Schaum's outline series. Material. Type. Book. Language English. Title. Schaum's . Schaum's Outline of Set Theory and Related Topics book. Read 3 reviews from the world's largest community for readers. This will be a revision of the fir.
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Schaum's Outline - Set Theory - Ebook download as PDF File .pdf) or read book online. Schaum's Outline of Theory and Problems of. SET THEORY. Copyright . Although we shall study sets as abstract entities, we now list ten examples of sets . Schaum's Outline of Set Theory and Related Topics by Seymour Lipschutz, You also get hundreds of examples, solved problems, and practice exercises to.
Accordingly, R is not a transitive relation. AnE cA. Suppose sets A and B are not comparable. Identity Law, Commutative Law 7. Associative opera- tions.
Details if other: Thanks for telling us about the problem. Return to Book Page. This will be a revision of the first edition published in and will introduce current topics such as Turning Machines and Computable Functions. Some outdated chapters on sets relating to new math will be removed.
Problems will be improved in the Ordinals, Cardinals, and Transfinite Induction chapters. Get A Copy. Paperback , pages. More Details Original Title. Other Editions 7.
Friend Reviews. To see what your friends thought of this book, please sign up. This question contains spoilers… view spoiler [how i can download? Lists with This Book. This book is not yet featured on Listopia. Community Reviews. Showing Rating details. Sort order. Feb 22, Kototama rated it liked it. Really clear introduction to set theory, with exercices and answers.
A lot of subjects are explained: However the book suffers from several shortcomings: Consider the six functions in Problem Consider the functions in Problem Which of the following is always true: State whether or not each of the following properties defines a one-one function: State whether or not each of the functions in Problem 54 is one-one. State whether or not each of the functions in Problem 55 is one-one.
State whether or not each of the functions in Problem 52 is one-one. The functions f-. State whether or not each of the following defines a product function and if it does, determine its domain and co-domain: Let g: Let f-.
Nine They are all one-one. Only g is one-one. The ordered pairs 2, 3 and 3, 2 are different. The points in the Cartesian plane shown in Fig. Ordered pairs can have the same first and second elements such as 1, 1 , 4, 4 and 5, 5. Although the notation a, b is also used to denote an open interval, the correct mean- ing will be clear from the context.
Remark 5. The product set of A and B consists of all ordered pairs a, b where a e A and b e B. It is denoted by AxB which reads "A cross B". The Cartesian plane shown in Fig. It is named after the mathematician Descartes who, in the seventeenth century, first investigated the set R x R. If set A has n elements and set B has m elements then the product set A x B has n times m elements, i.
If either A or B is the null set then A x B is also the null set. Lastly, if either A or B is infinite and the other is not empty, then A x B is infinite. Each point P represents an ordered pair a, b of real numbers. A vertical line through P meets the horizontal axis at a and a horizontal line through P meets the vertical axis at b, as in Fig. Here the elements of A are displayed on the horizontal axis and the elements of B are displayed on the vertical axis.
Notice that the vertical lines through the elements of A and the horizontal lines through the elements of B meet in 12 points. These points repre- sent AxB in the obvious way. The point P is the ordered pair c, y. Let N be the natural numbers 1, 2, 3, First, for each element aeA there is assigned an element in B.
Secondly, there is only one element in B which is assigned to each aeA. Property 1: Property 2: Specifi- cally, 3 e A and there is no ordered pair in which 3 is a first element. Let f x — x 2 define a function on the interval —2 - x — 4.
In 1 , the vertical line through 6 does not contain a point of the set; hence the set of points cannot be the graph of a function of A into B. In 2 , the vertical line through a contains two points of the set; hence this set of points cannot be the graph of a function of A into B.
Property 1 guarantees that each element in A will have an image, and Property 2 guarantees that the image is unique. In view of the correspondence between functions f: The geometrical implication of Definition 5. Analogously, the Cartesian product of n sets Ai,A 2 ,. Here an ordered w-tuple has the obvious intuitive meaning, that is, it consists of n elements, not necessarily distinct, in which one of them is designated as the first element, another as the second element, etc. In three dimensional Euclidean geometry each point represents an ordered triplet, i.
Find W x V. W X V consists of all ordered pairs a, b where a e W and beV. Find x and y. The vertical line through Pi crosses 6 on the A axis, and the horizontal line through Pi crosses i on the B axis; thus Pi corre- sponds to the ordered pair b,i. Draw light vertical lines through 1 and 4 on the horizontal axis, and horizontal lines through —2 and 3 on the vertical axis as shown in Fig. The rectangular area bordered by the four lines, together with three of its sides, represents the product set.
Shade the diagram as shown in Fig. Notice that the side of the rectangle which does not belong to the product set is shaded by a dotted line. Find AxBxC. By definition of intersection, y belongs to both B and to C. Since x e A and? Now let z,w be any element of A X J? It now follows that zeA and joeB, and zzA and weC. Since w belongs to both B and C, then weBnC. Find SxW n SxV. Let the function h: To each letter in S let the function g assign the letter which follows it in the alphabet.
State whether or not each of the following sets of ordered pairs is a function of V into V. Note first, by Definition 5. Hence ft is not a function of V into V. Although 2 appears as the first element in two ordered pairs, these two ordered pairs are equal.
State whether or not the set of points in each of the following coordinate diagrams of W x W is a function from W into W. Note first that a set of points on a coordinate diagram is a function provided that every vertical line contains one and only one point of the set. The fact that the horizontal line through c contains three points does not violate the properties of a function. The set g of points in the coordinate diagram of R x S in the figure at right is a function of R into S.
The points are 1,2 and 5,2. Geometrically, this set consists of the first elements of the points in g which lie below the horizontal line through 3. Let h be a set of points in the coordinate diagram of E x F which is a function of E into F.
Thus h is a one-one function. Hence h is an onto function. State whether or not each of the following sets of points displayed on a coordinate diagram of W x W is a function of W into W.
Note first, by Remark 5. Find the inverse function. To find the inverse function, permute, i. Find a? Sketch, by shading the appropriate area, each of the following product sets on a coordinate diagram offl xfl. Suppose sets V, W and Z have 3, 4 and 5 elements respectively. Which, if any, of the following is true?
Find the graph of g. Each of the following formulas defines a function of i2 into R. Plot the graph of each of these functions on a coordinate diagram of R XR , the Cartesian plane. State whether or not each of the following sets of ordered pairs is a function of W into W.
State whether or not the set of points in each of the following coordinate diagrams of V X V is a function of V into V. Let B — [—4, 4]. State whether or not each of the following sets of points displayed on a coordinate diagram of B X B is a function of B into B. Sketch, by shading the appropriate area, each of the following product sets on a coordinate diagram of R X. Each of the following formulas defines a function of j? Plot each of these functions on a coordinate diagram of R X R.
See Fig. Each has 60 elements. Both are true. The expression P x,y by itself shall be called an open sentence in two variables or, simply, an open sentence. Other examples of open sentences are as follows: It is also possible to have open sentences in one variable such as "x is in the United Nations", or in more than two variables such as "x times y equals z".
Then Ri is a relation since P a, b , i. Then R 2 is a relation. Then Ra is a relation. Furthermore, 3 R 3 12, 2 j? Then Rs is a relation. The open sentence P x,y , which reads "x is less than y", defines a relation in the rational numbers.
The open sentence "x is the husband of y" defines a relation from the set of men to the set of women. Some authors call the expression P x,y a relation. They then assume implicitly that the variables x and y range, respectively, over some sets A and B, i. We shall adhere to the previous terminology where P x,y is simply an open sentence and, hence, a relation consists of P x, y and two given sets A and B.
The graph of a relation R from A to B consists of those points on the coordinate diagram of A x B which belong to the solution set of R. Although Definition 6. Remark 6. Let set A have m elements and set B have n elements. Then there are 2 mn different relations from A to B, since Ax B, which has mn elements, has 2 mn different subsets. The inverse relation of R is i? Then R is called a reflexive relation if, for every a e A, a, a e R In other words, 72 is reflexive if every element in A is related to itself.
R is not a reflexive relation since 2, 2 does not belong to R. Notice that all ordered pairs a, a must belong to R in order for R to be reflexive. Let A be the set of triangles in the Euclidean plane. The relation R in A defined by the open sentence "x is similar to y" is a reflexive relation since every triangle is similar to itself.
Let R be the relation in the real numbers defined by the open sentence "x is less than y", i. Then J? Then R is called a symmetric relation if a, b e R implies b, a zR that is, if a is related to b then b is also related to a. Let A be the set of triangles in the Euclidean plane, and let R be the relation in A which is defined by the open sentence "x is similar to y". Then R is symmetric, since if triangle a is similar to triangle 6 then b is also similar to a.
Let R be the relation in the natural numbers N which is defined by "x divides y". Then R is not symmetric since 2 divides 4 but 4 does not divide 2. R Remark 6. Let N be the natural numbers and let R be the relation in N defined by "x divides y". Let cA be a family of sets, and let R be the relation in cA defined by "x is a subset of y". Let D denote the diagonal line of Ax A, i.
Let A be the set of people on earth. If a loves b and 6 loves c, it does not necessarily follow that a loves c.
Accordingly, R is not a transitive relation. In a later chapter we will more fully study equivalence relations in sets. Now we just give two examples of equivalence relations. Let R be the relation on A denned by "x is similar to y". Then, as proven in geometry, R is reflexive, sym- metric and transitive.
Thus R is an equivalence relation. The most important example of an equivalence relation is that of "equality". For any elements in any set: R is displayed on the coordinate diagram of R X R as shown in the figure at right. The domain of R is the closed interval [—3,3], and the range of R is the closed interval [-2, 2]. Then aeA is in the domain of R if, and only if, the vertical line through a contains a point of the graph of R. Also, beB is in the range of R if, and only if, the horizontal line through b contains a point of the graph of R.
Since every subset of A x B is a relation, a function is a special type of a relation. In fact, the terms "domain" and "range" appeared both in the discussion of functions and in the discussion of relations. In general, this problem is extremely difficult.
Here, we are only able to answer this question for very simple equations. Notice that R is a circle of radius 5 with center at the origin. Notice, further, that many vertical lines contain more than one point of R. In particular, 3, 4 e R and -5 3, —4 e R. Thus the relation R is not a R is displayed function. Notice that R is the upper half of a circle. Notice further that each vertical line contains one and only one point of R; hence R is a function. A X B is shaded R is displayed Fig.
Notice that R is a straight line and that every, vertical line contains one and only one point of R; thus R is a function. Furthermore, by solving for y in terms of x in the equation above, we obtain a formula that defines the function R: Each of the following open sentences defines a relation in the real numbers.
Sketch each relation on a coordinate diagram of R xR. In order to sketch a relation on the real numbers which is defined by an open sentence of the form CHAP. Then the relation, i. Sketch each relation on a coordinate diagram of RxR. The relation will consist of all the points in possibly one or more regions. Test one or more points in each region in order to determine whether or not all the points in the region belong to the relation.
The sketch of each of the above relations is as follows: The sketch of R on the coordinate diagram of R x R follows: What relationships, if any, exist between the domain and range of a relation R and the domain and range of i? Let R be the relation in the natural numbers N — 1,2,3,..
When is a relation R in a set A not reflexive? R is not reflexive if there is at least one element aeA such that a, a f. Is R reflexive? R is not reflexive since 3eW and 3,3 R.
What relationship is there, if any, between any reflexive relation R in A and Dt Solution: Every reflexive relation R in A must contain the "diagonal line".
In other words, D is a sub- set of R if R is reflexive. Each of the following open sentences defines a relation R in the natural numbers N. State whether or not each is a reflexive relation. Hence R is reflexive. Hence R is not reflexive. Consider the following relations in E: If a relation in E is reflexive, then 1,1 , 2,2 and 3,3 must belong to the relation. Therefore only R 3 and R 5 are reflexive. When is a relation R in a set A not symmetric? Is R symmetric? R is not symmetric, since 3 e V, 4e7, 3, 4 e R and 4, 3 f, R.
Is there any set A in which every relation in A is symmetric? If A is the null set or if A contains only one element, then every relation in A is symmetric. State whether or not each relation is symmetric. Thus R is not symmetric. Hence R is not symmetric. Hence R is sym- metric. Thus R is not sym- metric. Let i? Let a, 6 belong to RnR'. Since JS and i?
When is a relation R in a set A not anti-symmetric? Is R anti-symmetric? R is not anti-symmetric, since 1 e W, 2 e W, 1 2, 1, 2 e 2? Can a relation i2 in a set A be both symmetric and anti-symmetric?
Consider the following- relations in E: Give an example of a relation R in E such that R is neither sym- metric nor anti-symmetric. R is also not anti-symmetric since 1,2 e R and 2,1 e R. State whether or not each of the relations is anti-symmetric. Hence R is anti-symmetric. When is a relation in a set A not transitive? Is R transitive? State whether or not each relation is transitive.
R Hence R is not transitive. If a relation R is transitive, then its inverse relation R' 1 is also transitive. Let a, 6 and 6, c belong to ft -1 ; then c, b e R and b,a eR. We have shown that a,b z R' 1 , 6,c ei? Consider the following relations in W: Note first that a relation R in W is a function of W into W if and only if each aeW appears as the first element in one and only one ordered pair in R.
Let the relation R from A to B be sketched on the coordinate diagram of A x B, How could one determine geometrically whether or not R is, in fact, a function of A into B? If every vertical line contains exactly one point of R, then R is a function of A into B. Consider the following relations: Sketch Bx A and it! B X A is shaded R is plotted Fig. Thus R is reflexive. Thus 6, a belongs to R. Since a, b e R implies 6, a e R R is symmetric.
Since a, 6 e R and 6, c e R implies a, c e R R is transitive. Since R is reflexive, symmetric and transitive, R is, by definition, an equivalence relation. Let R and R' be relations in a set A. Prove each of the following two statements: Hence 6, a also belongs to R or R'.
Therefore RuR' is reflexive. Show that each of the following statements is false by giving a counter example, that is, an example for which it is not true. The cross-hatched area is RnR'. Thus RnR' is displayed in Fig. R and R' are sketched Fig. Write R as a set of ordered pairs. Sketch each relation on a coordinate diagram of R XR.
Find 2 the domain of R, 3 the range of R. Find 2 the domain of R, 3 the range of R, 4 R' 1. In CHAP. State whether each of the following statements is true or false. Assume R and R' are relations in a set A. Let L be the set of lines in the Euclidean plane and let R be the relation in L defined by "x is parallel to y". State whether or not R is 1 reflexive, 2 symmetric, 3 anti-symmetric, 4 transitive. Assume a line is parallel to itself.
Let L be the set of lines in the Euclidean plane and let R be the relation in L defined by "x is perpendicular to y". Let qA be a family of sets and let R be the relation in oA defined by "x is disjoint from y". Each of the following open sentences defines a relation in the natural numbers N. Consider the following relations in T: If X and Y are each permitted to be any of the above four sets, which of the sixteen relations are functions?
First plot P x,y on a Cartesian plane. Let A be any set. Let A be non-empty and let R be a transitive relation in A which does not contain any of the "diagonal elements" x,x e A X A; then R is not a function in A.
Consider the following relations in the real numbers: Consider each of the following sets of ordered pairs of real numbers, i. Let A be the set of people.
Each open sentence below defines a relation in A. For each of these relations find an open sentence, sometimes called the "inverse sentence", that defined the inverse relation. Let N be the natural numbers. Each open sentence below defines a relation in N. For each of the relations, find an open sentence which defines the inverse relation.
R is reflexive, symmetric and transitive, i. R is not anti-symmetric. R is only symmetric. The only relations which are functions are: The only reflexive relation in a set A which is a function is the relation which consists only of the ordered pairs on the "diagonal line" of A X A; it defines the identity function on A. Table 1 lists laws of sets, most of which have already been noted and proven in Chapter 2. One branch of mathematics investigates the theory of sets by studying those theorems that follow from these laws, i.
We will refer to the laws in Table 1 and their consequences as the algebra of sets. An A' - 8b. Although these concepts were essential to our original develop- ment of the theory of sets, they do not appear in investigating the algebra of sets. Statement 1. AcC algebra of sets, that is, we prove in Table 1. Other theorems and l. Reason 1. Distributive Law 2. Complement Law 3. Substitution 4. Identity Law 5. Substitution Reason 1.
Definition of subset 2. Substitution 3.
Associative Law 4. Substitution 5. This fact is extremely important in view of the following principle: If certain axioms imply their own duals, then the dual of any theorem that is a consequence of the axioms is also a consequence of the axioms. For, given any theorem and its proof, the dual of the theorem can be proven in the same way by using the dual of each step in the original proof. Thus the principle of duality applies to the algebra of sets.
Principle of Duality: Since to each element iel there is assigned a set A if we state Definition 7. Let I be the set of words in the English language, and let iel. Remark 7.
Specifically, the identity function i: These definitions can easily be extended, by induction, to a finite number of sets. These concepts are generalized in the following way.
Then U ie j Ai consists of those elements which belong to at least one Ai where ieJ. Specifically, Theorem 7. For any i,l? Furthermore, each 2? Notice that Wi and Ws are not dis- joint, but there is no contradiction since the sets are equal. This example is highly instructive. The reader should actually find the sets W 4 and then verify that they do define a partition of A. Definition 7. A relation R in a set A is an equivalence relation if: The reason that partitions and equivalence relations appear together is because of the Theorem 7.
In other words, an equivalence relation R in a set A partitions the set A by putting those elements which are related to each other in the same equivalence class.
The converse of the previous theorem is also true. Then R is an equivalence relation in A. Thus there is a one to one correspondence between all partitions of a set A and all equivalence relations in A. In the Euclidean plane, similarity of triangles is an equivalence relation. Thus all triangles in the plane are partitioned into disjoint sets in which similar triangles are elements of the same set.
Write the dual of each of the following: Prove the Right Distributive Law: Statement Reason 1. Commutative Law 2. Distributive Law 3. Commutative Law 3. Method 1. The dual of this theorem was proven in Problem 2. Hence the theorem is true by the Principle of Duality. Method 2. Commutative Law 4. By the Principle of Duality, the theorem is true since its dual was proven in Ex- ample 1.
Identity Law 2. Hypothesis 3. Right Distributive Law 5. Complement Law 6. Identity Law, Commutative Law 7. A'cB 7. Hence 1"! Let x belong to Bn U ieI A t. State whether or not each of the following families of sets is a partition of A. Prove Theorem 7. Since R is reflexive, i. Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject.
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