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4.2 preimage_of_set (read the 3 thms of sec.7)

# 4.2 preimage_of_set (read the 3 thms of sec.7) - 12...

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Unformatted text preview: 12 FUNDAMENTALS or SET THEORY [CFL I Rules according to which the elements of a set are partitioned into classes can be of the most varied sort. But of course all these rules cannot be entirely arbitrary. Let us assume, for example, that we should hke to partition all real numbers into classes by including the number b 1n the same class as the number a if, and only if, b > a. It is clear that no partition of the real numbers into classes can be obtained in this way because if b > a, i.e. if b must be included in the same class as a, then a .< b, i.e. the number a must not belong to the same class as 1). Moreover, Since (1-18 not larger than a, then a ought not belong in the class which'contamsit! We consider another example. We shall see whether or not it is possrble to partition all inhabitants of a given city into classes by putting .two persons into the same class if, and only if, they are acquaintances. It is clear that such a partition cannot be realized because if A is an acquaintance of B and B is an acquaintance of C , then this does not at all mean that A is an acquaintance of 0. Thus, if we put A into the same class as B and B into the same class as C, it may follow that two persons A and 0 who are not acquaintances are in the same class. We obtain an analogous result if we attempt to partition the points of the plane into classes so that those and only those points whose mutual distance does not exceed 1 are put into one class. . . . The examples introduced above point out those conditions whlch must be satisﬁed-by any rule if it is to realize a partition of the elements of a set ' classes. miset M be a set and let some pair (a, b) of elements of this set be “marked.” [Here the elements a and b are taken in a deﬁnite order, i.e. (a, b) and (b, a.) are two distinct pairs] If (a, b) is a “marked” pair, we shall say that the element a is related to b by the relation (,0 and we shall denotetlns fact by means of the symbol a (p b. For example, if we Wlsh to partition triangles into classes of triangles having the same area, thena (o b is to mean: “triangle a has the same area as triangle b.” We shall say. that the given relation is an equivalence relation if it possesses the following propertles: 1. Reﬁexivity: a (p a for any element a E M; 2. Symmetry: if a (p b, then necessarily b (,0 a; 3. Transitivity: if a go b and b (p c, then a (p c. . Obviously every partition of a given set into classes deﬁnes some equiva— lence relation among the elements of this set. . . In fact, if a (p b means “a belongs to the same class as b”, then this relation will be reﬂexive, symmetric and transitive, as is easy to verify. Conversely, if a «o b is an equivalence relation between the elements. of the set M, then putting into one class those and only those elements which are equivalent we obtain a partition of the set M into classes. . In fact, let K, be the class of elements in M which are equlvalent to a §7] MAPPINGS or SETS. FUNCTIONS 13 ﬁxed element a. By virtue of the property of reﬂexivity the element a itself belongs to the class K, . We shall show that two classes Ka and K1, either coincide or are disjoint. Let an element 6 belong simultaneously to Kl, and to K, , i.e. Cga aand mob. Then by Virtue of symmetry, a (,9 c, and by virtue of transitivity, (1) a go I). Now if x is an arbitrary element in K, , i.e. :1: <9 0., then by virtue of (1) and the transitivity property, a: go b, i.e. a; 6 K1, . Conversely, if y is an arbitrary element in Kb , i.e. if y (,0 b, then by virtue of relation (1) which can be written in the form b (,9 a (symmetry!) and the transitivity property, y (P a, i.e. y E K“ . Thus, two classes Ka and K 1, having at least one element in common coincide. We have in fact obtained a partition of the set M into classes according to the given equivalence relation. §7. Mappings of sets. General concept of function In analysis the concept of function is introduced in the following way. Let X be a set on the real line. We say that a function f is deﬁned on this set if to each number a: E X there is made to correspond a deﬁnite number y = f(x). In this connection X is said to be the domain of the given func— tion and Y, the set of all values assumed by this function, is called its range. Now if instead of sets of numbers we consider sets of a completely arbi- trary nature, we arrive at the most general concept of function, namely: Let M and N be two arbitrary sets; then we say that a function f is deﬁned on M and assumes its values in N if to each element a: E M there is made to correspond one and only one element in N. In the case of sets of an arbitrary nature instead of the term “function” we frequently use the term “mapping” and speak of a mapping of one set into another. If a is any element in M, the element b = f(a) in N which corresponds to it is called the image of the element a (under the mapping f). The set of all those elements in M whose image is a given element b E N is called the inverse image (or more precisely the complete inverse image) of the element b and is denoted by f 1(b). If A is any set in M, the set of all elements of the form {f(a) :a E A} is called the image of A and is denoted by f (A). In its turn, for eyery set B in N there is deﬁned its inverse image f1(B), namely f‘1(B) is the set of all those elements in M whose images belong to B. In this section we shall limit ourselves to the consideration of the most general properties of mappings. We shall use the following terminology. We say that f is a mapping of 14 FUNDAMENTALS 015‘ SET THEORY [cm I the set M onto the set N if f(M) = N; in the general case, 1.e. when f (M ) E N we say that f is a mapping ofM into N. We shall establish the following properties of mappings. THEOREM 1. The inverse image of the sum of two sets is equal to the sum of their inverse images: f"1(A U B) = f'1(A) U f'1(B)- Proof. Let the element to belong to the set f‘XA U B). This means that f(x) E A U B, is. f(rc) E A or ﬁx) E B. But then :1: belongs to at least one of the sets f‘1(A) and f‘1(B),- i.e. ‘11; E f'1(A) U f (B). Conversely, 1f a“. E f1(A) U f1(B), then 2: belongs to at least one of the sets 3" (A) and f1(B), i.e. f(zr) belongs to at least one of the sets 1A and B and conse— quently f(rc) E A U B, whence it follows that .r. E f‘ (A U B): , THEOREM 2. The inverse image of the intersection of two sets is equal to the intersection of their inverse images: PM fl B) = f“1(A)nf‘.1(B)- Proof. Ifrc E f_I(A H B), thenf(x) E A n B, i.e.f(m) E A andf(x) E B; » . 1 nsequently, a: E f’l(A) ands: E f1(B),1.e.x E f‘1(A) ﬁf— (B). 00Conversely, if a: E flu-1) [if—1(3), i.e. a; E f1(A) and a; E f1(B), then f(x) E A and ﬁre) E B, or in other words f(rc) E A n B. Consequently a: E f‘1(A n B). . . A Theorems 1 and 2 remain valid also for the sum and intersection of an arbitrary ﬁnite or inﬁnite number of sets. . Thus, if in N some system of sets closed with respect to the operations of addition and taking intersections is selected, then their inverse images in M 'form a system which is likewise closed with respect to these operations. THEOREM 3. Thelimage of the sum of two sets is equal to the sum of their images: f(A U B) = f(A) U f(B)- Proof. If y E f(A U B), then y = f(x), Where :1; belongs tO at least one of the sets A and B. Consequently, y = f(x) E f(A) U f(B). Conversely, if y E f(A) U f(B) then y = f(x), where :1: belongs to at least one of the sets. A and B, i.e. :1: E A U B and consequently y = f(x) E f(A U B). We note that in general the image of the intersection of two sets does not coincide with the intersection of their images. For example, let the mapping considered be the projection of the plane onto the ag-axis. Then the seg- ments <ll mm {C ’E l/\ I/\ 0 1; y 0 1; y do not intersect, but at the same time their images coincide, 0,, 1 §7] MAPPINGS OF sn'rs. FUNCTIONS 15 The concept of mapping of sets is closely related to the concept of par- titioning considered in the preceding sectiOn. ’ Let f be a mapping of the set A into the set B. If we collect into one class all those elements in A whose images in B coincide, we obviously Obtain a partition of the set A. Conversely, let us consider an arbitrary set A and an arbitrary partitioning of A into classes. Let B be the totality Of those classes into which the set A is.partitioned. If we correspond to each element- a E A that class (i.e. that element in B) to which (1 belongs, we obtain a mapping of the set A into the set B. EXAMPLES. 1. Consider the projection Of the xy-plane onto the x-axis. The inverse images of the points of the x-axis are vertical lines. Conse- quently to this mapping there corresponds a partitioning of the plane into parallel straight lines. 2. Subdivide all the points of three—dimensional space into classes by combining into one class the points which are equidistant from the origin of coordinates. Thus, every class can be represented by a sphere of some radius. A realization of the totality of all these classes is the set of all points lying on the ray (0, on). So, to the partitioning of three-dimensional space into concentric spheres there corresponds the mapping of this space onto a half-line. 3. Combine all real numbers having the same fractional part into one class. The mapping corresponding to this partition is represented by the mapping of the real line onto a circle. ...
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