AG1 / dixon_m_kurdachenko_l_subbotin_i_algebra_and_number_theory_a

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can list all pairs selected by this correspondence. Taking into account that a set of pairs is connected to the Cartesian product, we make the following definition.

1.2.1. Definition. Let A and B be sets. A subset <I> ofthe Cartesian product A x B is called a correspondence (more precisely, a binary correspondence) between A and B, or a correspondence from A to B. If A :::;= B, then the correspondence will be called a binary relation on the set A. If a = (x, y) E <1>, then we say that the elements x andy (in this fixed order) correspond to each other at a.

Often we will change the notation (x, y) E <I> to an equivalent but more natural notation x<I>y. The element x is called the projection of a on A, and the element y is called the projection of a on B. We can formalize these expressions by setting x = prAa andy= pr8a. We also define prA<l> = {prAa I a E <t>}, pr8 <1> = {pr8a I a E <1>}.

For example, let A be the set of all points in the plane and let B be the set of all lines in the plane. We recall that a point P is incident with the line A if P belongs to A and this incidence relation determines a correspondence between the set of points in the plane and the set of lines in this plane. All the lines that go through the point P correspond to P, and all points of the line A correspond to A. In this example, an unbounded set of elements of B corresponds to every element of the set A. In general, a fixed element of A can correspond to many, one, or no elements of B.

Here is another example. Let

A= {1, 2, 3, 4, 5}, B ={a, b, c, d}, and let

<I>= {(1, a), (1, c), (2, b), (2, d), (4, a), (5, c)}.

Then we can describe the correspondence <I> with the help of the following simple table:

Here all the elements of the set A correspond to the columns, while the elements of B correspond to the rows, the elements of the set A x B correspond to the points situated on the intersections of columns and rows. The points inside the circles (denoted by 0) correspond to the elements of <1>.

When A = B = JR, we come to a particularly important case since it arises in numerous applications. In this case, A x B = JR2 is the real plane. We can think of a binary relation on lR as a set of points in the plane. For example, the relation

<1> = { (x, y) I x 2 + l = 1}

10 ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH

can be illustrated as a circle of radius 1, with center at the origin. Another very useful example is the relation "less than or equal to," denoted as usual by :::;, on the set lR of all real numbers.

Next, let n be a positive integer. The relation congruence modulo n on Z is defined as follows. The elements a, b E Z are said to be congruent modulo n if a = b + kn for some k E Z (thus n divides a -b). In this case, we will write a = b (mod n). We will consider this relation in detail later.

Very often we can define a correspondence between A and B with the help of some property P(x, y), which connects the element x of A with the element y of B as follows:

<I>= {(x, y) I (x, y) E Ax Band P(x, y) is valid}.

This is one of the commonest ways of defining a correspondence.

1.2.2. Definition. Let A and B be sets and <I> be a correspondence from A to

A function or mapping f from a set A to a set B is a triple (A, B, <l>) where <I> is a functional correspondence from A to B. The set A is called the definitional domain or domain of definition of the mapping f; the set B is called the domain of values or value area of the mapping f; afunctional correspondence <I> is called the graph of the mapping f. We will write <I> = Gr(f). In short, A is called the domain off and B is called the codomain off.

If f is a mapping from A to B, then we will denote this symbolically by f: A---+ B.

Condition (F 1) implies that prA <I> = A. Thus, together with condition (F 2) this means that the mapping f associates a uniquely determined element b E B with every element a EA.

1.2.3. Definition. Let f : A ---+ B be a mapping and let a E A. The unique element b E B such that (a, b) E Gr(f) is called the image of a (relative to f) and denoted by f(a).

In some branches of mathematics, particularly in some algebraic theories, a right-side notation for the image of an element is commonly used; namely, instead of f(a) one uses af. However, in the majority of cases the left-sided notation is generally used and accepted. Taking this into account, we will also employ the left-sided notation for the image of an element.

Every element a E A has one and only one image relative to f.

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1.2.4. Definition. Let f : A ~ B be a mapping. If U s; A, then put

f(U) = {f(a) I a E U}.

This set f(U) is called the image of U (relative to f). The image f(A) of the whole set A is called the image of the mapping f and is denoted by lm f.

1.2.5. Definition. Let f: A ~ B be a mapping. If b = f(a) for some element a E A, then a is called a preimage of b (relative to f). If V s; B, then put

f- 1 (V) ={a E A I f(a) E V}.

The set f- 1(V) is called the preimage of the set V (relative to f). If V = {b} then instead of f- 1({b}) we will write f- 1(b).

Note that in contrast to the image, the element b E B can have many preimages and may not have any.

We observe that if A = 0, there is just one mapping A ~ B, namely, the empty mapping in which there is no element to which an image is to be assigned. This might seem strange, but the definition of a mapping justifies this concept. Note that this is true even if B is also empty. By contrast, if B = 0 but A is not empty, then there is no mapping from A to B. Generally we will only consider situations when both of the sets A and B are nonempty.

1.2.6. Definition. The mappings f: A~ B and g: C ~ Dare said to be equal if A= C, B = D and f(a) = g(a) for each element a EA.

We emphasize that if the mappings f and g have different codomains, they are not equal even if their domains are equal and f(a) = g(a) for each element

aEA.

1.2.7.Definition. Let f: A~ B be a mapping.

{i) A mapping f is said to be injective (or one-to-one) if every pair ofdistinct elements of A have distinct images.

(ii)A mapping f is said to be surjective (or onto) iflm f = B.

(iii)A mapping f is said to be bijective if it is injective and surjective. In this case f is a one-to-one correspondence.

The following assertion is quite easy to deduce from the definitions and its proof is left to the reader.

1.2.8.Proposition. Let f : A ~ B be a mapping. Then

(i)f is injective if and only if every element of B has at most one preimage;

12ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH

(ii)f is surjective if and only ifevery element of B has at least one preimage;

(iii)f is bijective if and only if every element of B has exactly one preimage.

(i)Iff is injective, then IAI :S IBI.

(ii)Iff is surjective, then lA I ~ IBI.

(iii)Iff is bijective, then lA I = IBI.

These assertions are quite easy to prove and are left to the reader.

(i)Iff is injective, then f is bijective.

(ii)Iff is surjective, then f is bijective.

!B(B), which associates with each subset U, of A, its image f(U). We will

(ii) If X~ U ~A, then f(X) ~ f(U).

These assertions are very easy to prove and the proofs are omitted, but the method of proof is similar to that given in the proof of Theorem 1.2.12 below. Note that in (iv) the symbol ~ cannot be replaced by the symbol =, as we

(i)IfY, v ~ B, then J-'crw) = J-'cnv-'cv).

(ii)IfY ~ V ~ B, then f-'(n ~ f- 1(V).

(iii)IfY, V ~ B, then f-'(n U f- 1(V) = f- 1(Y U V).

(iv)IfY, v ~ B, then !-'en n f- 1(V) = f- 1(Y n V).

Proof. (i)Leta E f- 1(Y\V).Thenf(a) E Y\V,sothatf(a) E Yandf(a) ~ V.

It follows that a E !-'en and a~ f- 1(V), so that a E f-'cn\f- 1(V). To prove the reverse inclusion we repeat the same arguments in the opposite order.

Assertion (ii) is straightforward to prove.

(iii) We first show that f- 1(Y U V) ~ f- 1en U f- 1(V) and to this end, let a E !-'en u f- 1(V). Then a E f-'cn ora E f- 1(V) and hence f(a) E Y or f(a) E V. Thus, in any case, f(a) E Y U V, so that a E f- 1(Y U V). We can

use similar arguments to obtain the reverse inclusion that f- 1en U f- 1(V) ~

t-'cr u V).

Similar arguments can be used to justify (iv).

for each element c E C, is called an identical embedding or a canonical injection.

For example, a canonical injection is the restriction of the corresponding identity mapping.

Let A and B be sets. The set of all mappings from A to B is denoted by BA.

14 ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH

Assume that both sets A and Bare finite, say IAI = k and IBI = n. We suppose that A = {a 1, ... , ak} and that f: A ~ B is a mapping. Iff E BA then

such that f(aj) =I= g(aj). It follows that <t>(f) =1= <t>(g) and hence <t> is injective. Furthermore, let (b 1 , ... , bk) be an arbitrary k-tuple consisting of elements of B. Then we can define a mapping h : A ~ B by h(aJ) = b 1, ••• , h(ak) = bk and

BA for the set of mappings from A to B. We shall also use this notation for infinite sets.

is called the characteristic function of the subset B.

1.2.16. Theorem. Let A be a set. Then the mapping B r---+ XB is a bijection from the Boolean IB(A) to the set {0, l}A.

Proof. By definition, every characteristic function is an element of the set {0, 1}A.

distinct subsets of A. Then, by Definition 1.1.1, either there is an element dE D such that d rf- E, or there is an element e E E such that e rf- D. In the first case, xv(d) = 1 and xdd) = 0. In the second case we have XE(e) = 1 and xv(e) = 0.

For every set A there is an injective mapping from A to IB(A). For example, a r---+ {a} for each a E A. However, the following result is also valid and has great implications.

1.2.18. Theorem (Cantor). Let A be a set. There is no surjective mapping from A onto IB(A).

1.2.19. Definition. A set A is called countable if there exists a bijective mapping f : N ----+ A. If an infinite set is not countable then it is said to be uncountable.

In the case when A is countable we often write an = f(n) for each n EN. Then

A = {a,, az, ... , an, ... } = {an I n E N}.

In other words, the elements of a countable set can be indexed (or numbered) by the set of all positive integers. Now we discuss some important properties of countable sets.

1.2.20. Theorem.

Proof.

(i) Let A be an infinite set so that, in particular, A is not empty and choose a, EA. The subset A\{a1} is also not empty, therefore we can choose an element a2 in this subset. Since A is infinite, A\{a 1, a 2 } =1= 0, so that we can choose an element a3 in this subset and so on. This process cannot terminate after a finite number of steps because A is infinite. Hence A contains the infinite subset {an I n E N}, which is countable.

(ii) Let A = {an 1 n E N}. Then there is a least positive integer k(l) such that ak(l) E B and we put h = ak(l)· There is a least positive integer k(2) such that ak(2) E B\{b, }. Put bz = ak(2)• and so on. If after finitely many steps this process terminates then the subset B is finite. If the process does not terminate then all the elements of B will be indexed by positive whole numbers.

(iii) We can list all the elements of the Cartesian product N x N with the help of the following infinite table.

16 ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH

Notice that if n is a natural number and if the pair (k, t) lies on the nth diagonal (where the diagonal stretches from the bottom left to the top right) then k + t = n + 1. Let the set Dn denote the nth diagonal so that

Dn = {(k, t) I k + t = n + 1}.

We now list the elements in this table by listing them as they occur on their respective diagonal as follows:

(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), ... , (1, n), (2, n- 1), ... , (n, 1), ...

"-..--'' - ... -''----..----'

D1 D2 D3 Dn+l

In this way we obtain a listing of the elements of N x N and it follows that this set is countable.

Assertion (ii) of Theorem 1.2.20 implies that the set of all positive integers in some sense is the least infinite set.

1.2.21. Corollary. Let A and B be sets. If A is countable and there is an injective mapping f : B - + A, then B is finite or countable.

n E N. Then A is finite or countable.

Proof. The set A1 is finite or countable, so that A1 = {an In E I:J}, where either I: 1 = N or I: 1 = {1, 2, ... , k1} for some k1 E N. We introduce a double indexing of the elements of A by setting b1n =an, for all n E I:J. Then, by Theorem 1.2.20, A2\A1 is finite or countable, and therefore A2\A1 ={an In E I:2} where either I:2 =Nor I:1 = {1, 2, ... , k2}, for some k2 EN. For this set also we use a double index notation for its elements by setting b2n =an, for all n E I:2. Similarly, we index the set A3\(A 2 U A1) and so on. Finally, we see that A= {bij I (i, j) E M} where M is a certain subset of the Cartesian product N x N.

It is clear, from this construction, that the mapping b;1 ~ (i, j) is injective and Corollary 1.2.21 and Theorem 1.2.20 together imply the result.

Since Z = UnEN{n- 1, -n + 1} we establish the following fact.

1.2.23.Corollary. The set Z is countable.

1.2.24.Corollary. The set Ql is countable.

To see this, for each natural number n, put An= {k/t llkl + ltl = n}. Then every subset An is finite and Ql = UnENAn, so that we can apply Corollary 1.2.22.

The implication that we make here is that all countable sets have the same "size" and, in particular, the sets N, Z, and Q have the same "size."

On the other hand, Theorem 1.2.18 implies that the set 113 (N) is not countable and Theorem 1.2.16 shows that the set {0, l}N is not countable. Each element of this set can be represented as a countable sequence with terms 0 and I. With each such sequence (a 1, a2, ... , an, ... ) we can associate the decimal representation of a real number 0 · a,a2 ... an ... from the segment [0, I]. It follows from this that the set [0, 1], and hence the set of all real numbers, is uncountable.

Corollary 1.2.9 shows that to establish the fact that two finite sets have the same "number" of elements there is no need to count these elements. It is sufficient to establish the existence a of bijective mapping between these sets. This idea is the main origin for the abstract notion of a number. Extending this observation to arbitrary sets we arrive at the concept of the cardinality of a set.

1.2.25. Definition. Two sets A and Bare called equipollent, ifthere exists a bijective mapping f : A ---+ B. We will denote this fact by IAI = IBI, the cardinal number of A.

If A and B are finite sets, then the fact that A and B are equipollent means that these sets have the same number of elements. Therefore Cantor introduced the general concept of a cardinal number as a common property of equipollent sets. Two sets A, B have the same cardinality if IAI = IBI. On the other hand, every infinite set has the property that it contains a proper subset of the same cardinality. This property is not enjoyed by finite sets. Now we can establish a method of ordering the cardinal numbers.

1.2.26. Definition. Let A and B be sets. We say that the cardinal number of A is less than or equal to the cardinal number of B (symbolically IA I .:::: IB 1), if there

is an injective mapping f : A B.

In this sense, Theorem 1.2.20 shows that the cardinal number corresponding to a countable set is the smallest one among the infinite cardinal numbers. Theorem 1.2.18 shows that there are infinitely many different infinite cardinal numbers. We are not going to delve deeply into the arithmetic of cardinal numbers, even though this is a very exciting branch of set theory. However, we present the following important result which, although very natural, is surprisingly difficult to prove. We provide a brief proof, with some details omitted.

1.2.27. Theorem (Cantor-Bernstein). Let A and B be sets and suppose that IAI .::::

IBI and IBI.:::: IAI. Then IAI = IBI.

Proof. By writing A1 =A x {0} and B1 = B x {1} and noting that A1 n B1 = 0 we may suppose, without loss of generality, that A n B = 0. Let f : A B and g : B ---+ A be injective mappings. Consider an arbitrary element a E A. Then either a E lmg and in this case a= g(b) for some element bE B, or

18 ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH

a ¢. Im g. Since g is injective, in the first case, the element b satisfying the equation a= g(b) is unique. Similarly, either b = f(aJ) for some unique a, E A, orb¢. Im f. For an element x E A we define a sequence (xn)nEN as follows. Put xo = x, and suppose that, for some n ::: 0, we have already defined the element Xn. If n is even, then we define Xn+ 1 to be the unique element of the set B such that Xn = g(xn+1). If such an element does not exist, the sequence is terminated at Xn. If n is odd, then we define Xn+l to be the unique element of the set A such that Xn = f(xn+J), with the proviso that the sequence terminates in Xn should such an element Xn+l not exist. Only the following two cases are possible:

1. There is an integer n such that the element Xn+ 1 does not exist. In this case we call the integer n the depth of the element x.

2. The sequence (xn)n:c:O is infinite. In this case, it is possible that the set {xn : n::: 0} is itself finite, as in the case, for example, when a= g(b) and b = f(a). In any case we will say that the element x has infinite depth.

We obtain the following three subsets of A:

the subset AE consisting of the elements of finite even depth; the subset Ao consisting of the elements of finite odd depth; the subset A 00 consisting of the elements of infinite depth.

We define also similar subsets B£, Bo, and B00 in the set B. From this con-

It is not hard to prove that h is a bijective mapping and this now proves the theorem.

As an illustration of how the language of sets and mappings may be employed in describing information systems, we consider briefly the concept of automata. An automaton is a theoretical device, which is the basic model of a digital computer. It consists of an input tape, an output tape, and a "head," which is able to read symbols on the input tape and print symbols on the output tape. At any instant, the system is in one of a number of states. When the automata reads a symbol on the input tape, it goes to another state and writes a symbol on the output tape.

To make this idea precise we define an automaton A to be a 5-tuple

(/, 0, S, v, a),

where I and 0 are the respective sets of input and output symbols, S is the set of states,

v:/xS----+0