Unformatted text preview: Module 2: Language of Mathematics Theme 1: Sets A set is a collection of objects. We describe a set by listing all of its elements (if this set is finite and not too big) or by specifying a property that uniquely identifies it. Example 1: The set of all decimal digits is But to define a set of all even positive integers we write: where is a natural number The last definition can be also written in another form, namely: is an even natural number In the rest of this course, we shall either write This notation is called the set builder. Let be a set such that elements property describing or property describing , where or should be read as "such as". Both are used in discrete math, however, we prefer the former. belong to it. We shall write if is an element of . If does not belong to we denote it as . Uppercase letters are usually used to denote sets. Some letters are reserved for often used sets such as the set of natural numbers (i.e., set of all counting numbers), the set of integers (i.e., positive and negative natural numbers together with zero), and the set of rational numbers which are ratios of integers, that is, The set We shall write Example 2: The set subset of that is not an element of Two sets and is said to be a subset of to indicate that .A . set with no elements is called the empty (or null) set and is denoted as . if and only if every element of is a subset of . . Actually, in this case or ). are is also an element of is a subset of is a proper , and we write it as . By proper we mean that there exits at least one element of (in our example such elements are , or are equal if and only if they have the same elements. We will subsequently and make this statement more precise. For example, the sets equal since order does not matter for sets. When the sets are finite and small, one can verify this 1 by listing all elements of the sets and comparing them. However, when sets are defined by the set builder it is sometimes harder to decide whether two sets are equal or not. For example, is the set (i.e., solutions of this weird looking equation , where equal to ? Therefore, we introduce another equivalent definition: and whenever , then if whenever ) , then . The last statement can be written as follows: if and only if (To see this, one can think of two real numbers show that sets. If is, is finite, then the number of elements of and (1) and that are equal; to prove this fact it suffices to .) This equivalence is very useful when proving some theorems regarding is called its cardinality and denoted as , that number of elements in A A set is said to be infinite if it has an infinite number of elements. For example, the cardinality of is , while is an infinite set. , then there are The set of all subsets of a given set Example 3: If is called the power set and denoted . subsets of , namely: Thus the cardinality of is Theorem 1. If Proof:1 The set if ܽ . . Now, we shall prove our first theorem about sets. , then ȴ has elements and we can name them any way we want. For example, , say , contains some elements form . Any subset of elements or better we can associate with every element . We can list these which is set to be an indicator of and zero otherwise. More formally, for every subset we construct an indicator belongs to ; other- with understanding that if and only if the th element of wise we set . For example, for ȴ an element of (i.e., ܽ ) while , the identifier of is since is not , that is, ܾ and ܿ . Observe that every set of desired cardinality of . Since can take only two values, and there are possibilities the total number of indicators is , which is the cardinality of . This completes the proof. has a unique indicator ܽ . Thus counting the number of indicators will give us the The Cartesian product of two sets where and and , denoted by , is the set of ordered pairs , that is, 1 2 and This proof can be omitted in the first reading. U A B A B AB U A B B U AB U AB B = UB Figure 1: Venn diagrams for the union, intersection, difference, and complementary set. Example 4: If and , then of In general, we can consider Cartesian products of three, four, or sets. If , then an element is called an -tuple. We now introduce set operations. Let intersection , and the difference and be two sets. We define the union , the , respectively, as follows: Example 5: Let and or and and , then We say that sets. 3 and are disjoint if , that is, there is no element that belongs to both Sometimes we deal with sets that are subsets of a (master) set . We will call such a set the universal set or the universe. One defines the complement of the set , denoted as , as diagrams as shown in Figure 1, which is self-explanatory. When is, and . We can represent visually the union, intersection, difference, and complementary set using Venn are disjoint, the cardinality of (provided is the sum of cardinalities of and and , that ). This identity is not true when are not disjoint, yielding since the intersection part would be counted twice!. To avoid this, we must subtract The above property is called the principle of inclusion-exclusion. An astute reader may want to generalize this to three and more sets. For example, consider the sets from Example 5. Note that , while , and , thus . We have already observed some relationships between set operations. For example, if , then , but . There are more to discover. We list these identities in Table 1. Table 1: Set Identities Identity Name Identity Laws Domination laws Idempotent laws Complementation laws Commutative laws Associative laws Distributive laws De Morgan's laws We will prove several of these identities, using different methods. We are not yet ready to use sophisticated proof techniques, but we will be able to use either Venn diagrams or the the principle 4 expressed in (1) (i.e., to prove that two sets are equal it suffices to show that one set is a subset of the other and vice versa). The reader may want to use Venn's diagram to verify all the identities of Table 1. Example 6: Let us prove one of the identities, say De Morgan's law, showing that and . First suppose or which implies that . Hence, or . (observe this by drawing the Venn diagram or referring to the logical de Morgan laws discussed in Module 1). Thus, Hence Suppose now that , that is, which implies or . This further implies that . This shows that or . , and therefore . This proves , and completes the proof of the De Morgan law. Exercise 2A: Using the same arguments as above prove the complementation law. 5 Theme 2: Relations In Theme 1 we defined the Cartesian product of two sets ordered pairs such that and and , denoted as , as the set of . We can use this to define a (binary) relation if it is a subset of the Cartesian product We say that is a binary relation from , then we write to . . If and say that and is related to . . Define the relation as Example 7: Let divides where by "divides" we mean with zero remainder. Then We now define two important sets for a relation, namely, its domain and its range. The domain of is defined as and for some while the range of is the set and for some In words, the domain of is composed of all all for which there is such that . The set of . while the range is . such that there exists is the range of Example 8: In Example 7 the domain of that domain of is a subset of is the set . Observe while the range is a subset of There are several important properties that are used to classify relations on sets. Let relation on the set . We say that: ; be a is reflexive if for every is symmetric if implies for all ; is antisymmetric if and implies that is transitive if and implies for all ; . Example 9: Consider the relation is: and let . Observe that not reflexive since ; not symmetric since for example but 6 ; not antisymmetric since and but not transitive since ; and , but . . The On the other hand, consider this relation relation is reflexive, transitive, but not symmetric, however, its antisymmetric, as easy to check. Exercise 2B: Let . Is the following relation reflexive, symmetric, antisymmetric, or transitive? Relations are used in mathematics and in computer science to generalize and make more rigorous certain commonly acceptable notion. For example, " " is a relation that defines equality between elements. Example 10: Let be the set of rational numbers, that is, ratios of integers. Then means that has the same value as . For example, . The relation partitions the set of all rational numbers into subsets such every subset contains all numbers that are equal. Observe that is reflexive, symmetric and transitive. Indeed, , is the same as , and finally if and , then . Such relations are called equivalence relations and they play important role in mathematics and computer science. A relation set that is reflexive, symmetric, and transitive is called an equivalence relation on the divides (actually, partitions) the set of all rational . As we have seen above, the relation numbers into disjoint sets that cover the the whole set of rational numbers. Let us generalize this. For an equivalence relation we define the equivalence set for any denoted by as follows In words, the whole set sets). is the set of all elements such that and are related by . This is a special set, and are disjoint and is the sum of disjoint equivalence as we formally explain below. Informally, the two different equivalence sets is partitioned into disjoint equivalence sets (i.e., More formally, observe that if , hence , then . Thus , then . Indeed, let . We shall prove that and from and transitivity we conclude that , hence . , as needed. In a similar manner we can prove that (by definition of which proves that Clearly if logically equivalent, manner: If into disjoint subsets other words, the set ). The latter can be expressed in a different, but (this is an example of counterpositive can be partitioned . In belongs to exactly one equivalence class , then argument discussed in Module 1: Basic Logic). We should conclude that the set such that every element of 7 is a partition of discuss. . There is one important example of equivalence classes, called congruence class, that we must Example 11: Fix a number that is a positive integer (i.e., a natural number). Let integers. We define a relation on be the set of all such that since by if is divisible by , that is, there is an integer . We will write for , or more often a relation is also called congruence modulo n. For example, since , but . Such is not divisible by . It is not difficult to prove that . If is reflexive, symmetric and transitive. Indeed, , then let and . That is, since and implies . Then . Finally, since , hence Since . is an equivalence relation, we can define equivalence classes which are called congruence classes. From the definition we know that for some Example 12: Congruence classes modulo are In words, an integer belongs to one of the above classes because when dividing by the remainder is either or or or or , but nothing more than this. We have seen before that the relation the same with well known was generalized to the equivalence relations. Let us do if and . relation. Notice that it defines an order among of real number. Let now Clearly, this relation is reflexive and transitive because and if unless and , then . It is definitely not symmetric since doe snot imply and . Actually, it is easy to see that it is antisymmetric since if , then . We call such relations partial orders. More precisely, a relation reflexive, antisymmetric, and transitive. Example 13: Let if divides (evenly) for we saw in Example 11. It is not symmetric (indeed, on a set is partial order if is . This relation is reflexive and transitive as divides but not the other way around). Is it antisymmetric? Let divides and divides , that is, there are integers and such that 8 and . That is, , hence . Since we must have . Thus is antisymmetric. total order) since for that and A reflexive, antisymmetric, and transitive relation is called partial order (not just an order or it may happen that neither . If for all nor . In Example 13 we see we have either or , then defines a total order. For example, the usual ordering of real numbers defines a total ordering, but pairs of real numbers in a plane define only a partial order. 9 Theme 3: Functions Functions are one of the most important concepts in mathematics. They are also special kinds of relations. Recall that a relation from also that the domain of is the set of to is a subset of the Cartesian product . Recall such that there exists related to through , that is, there is . For relations it is not important that for every related to by . Moreover, it is from legitimate to have two 's, say ݽ and ݾ such that ݽ and ݾ for some . These two properties are eliminated in the definition of a function. More formally, we define a function denoted as to as a relation from is ݾ to ; having two additional properties: 1. The domain of 2. If ݽ and , then ݽ ݾ . The last item means that if there is an such that it is related to ݽ and ݾ , then ݽ must be equal to ݾ . In other words, there is no that has two different values of related to it. We shall use lowercase letters , , , etc. to denote functions. Furthermore, when we shall write it as ܵ. Finally, we will also use another standard notation for functions, namely: Functions are also called mappings or transformations. The second property of the function definition is very important, so we characterize it in another way. Consider a relation on and . Define Observe that ʴܵ is a set. It may be empty, may contain one element or many elements. When is a function, then ʴܵ is not empty for every defined as and in fact it contains exactly one element that is denoted as for a function ܵ for some called an image of . More generally, the image of is . In other words, is a subset of Figure 2 the image of is for which there is such that ܵ . For example in to Example 14: (a) Consider the relation . It is a function since every from has exactly one image in . In fact, and , and . Figure 2 shows a graphical representation of this function. is not a function since have the same (b) The relation image . 10 1 2 3 a b c d Figure 2: The function defined in Example 14. 8 8 6 6 4 4 2 2 -3 -2 -1 1 x 2 3 -3 -2 -1 1 x 2 3 (a) Figure 3: Plots of two functions: (a) ܵ (b) ; (b) ܵ . Functions are often represented by mathematical formulas. For example, we can write ܵ for every real , or more formally ܵ or ܵ. To visualize such functions we often graph them in the coordinates where Example 15: In Figure 3 we draw the functions are defined on the set of reals hence the range of which is the domain for both functions. Since and . Both functions and is the set of nonnegative reals while for and it is the set of positive reals. That is, . Exercise 2C: What is the image of for ܵ function ܵ ? What about the image of over the ? 11 There are some functions occurring so often in computer science that we must briefly discuss them here. The first function, the modulus operator, we already studied in Example 11. We say that is equal to the remainder when is divided by (we, of course, implicitly assume that , i.e., and are integers). We recall that and . is equivalent to used before. For example, We shall write this function as ܵ with the understanding that is the remainder of the division range of to the set . The domain of such a function is the set of integers, while the image (or range) is the set of natural numbers. In fact, we can restrict the between and . because the remainder of any division by must be an integer , then The other important and often used functions are the floor and ceiling of a real number. Let the greatest integer less than or equal to the least integer greater than or equal to For example, Finally, we introduce some classes of functions as we did with relations. Consider the function ܵ shown in Figure 3(a). We have , that is, there are two values of that to is one-to-one or injective if there are mapped into the same value of (or with the same image). This is an example of a function that is not one-to-one or injective. We say that a function are ܽ ܾ for each from ܾ there is at most one such that if ܽ ܾ , then ܽ . In other words, for one-to-one function with ܵ . The function in Figure 3(b) is one-to-one, as easy to see. Example 16: Consider from to . This function is injective. How to know weather a function is one-to-ne or not? We provide some conditions below. We first introduce increasing and decreasing functions. A function decreasing) if ܵ ( ܵ ) whenever for all is increasing (non. For example, the 12 function ܵ is increasing in the domain is decreasing (non-increasing) if ܵ will be. ( ܵ (cf. Figure 3(b)). Similarly, a function ) whenever for all . For an increasing (decreasing) function the bigger the value of The function ܵ is, the bigger (smaller) the value of . However, it is a decreasing function for all negative reals and increasing in the set of all positive reals. In Example 16 we have is said to be onto Example 17: Let numbers. Clearly, ܵ plotted in Figure 3(a) is neither increasing or decreasing in the domain . A function be such that ܵ , thus it is onto from to such that ʵ or surjective function. , where . But if we define is the set of positive real with , then such a function is not surjective. that is both injective and surjective is called a bijection. The function in as A function inverse function Example 16 is a bijection while the function in Example 17 is not. For a bijection we can define an that is, and switch their roles. Observe that we do not need to have bijection in order to define the inverse since: (i) the domain of the inverse function is every there must be such that and this is guaranteed by the requirement that Example 18. Let restrict the domain range, that is, is ݵ; (ii) There must be only one such that is one-to-one function. ܵ and by the definition of a function, for ݵ be such that . Now ܵ . This is not a one-to-one function. Let us to the set of nonnegative reals and we do the same with the has an inverse function defined on which . Finally, we define the composition of two functions. Let and Then for every Example 19: Let we find ܵ , but for such we compute and is denoted as function is called the composition of and ܵ. The resulting . Then Example 20: Let ܵ Ҵܵ and ܵ . The composition ܵ Ҵܵ . 13 Theme 4: Sequences, Sums, and Products Sequences are special functions whose domain is the set of natural numbers , that is, is a sequence. We shall write or ҵ to denote an element of a sequence, where the letter in can be replaced by any other letter, say or . Since a sequence is a set we often write it as Ҿ or simply . Example 21: Let for . That is, the sequence starts with If , then the sequence begins with Finally, looks like by selecting only some terms. For We can create another sequence from a given sequence example, we can take very second term of the sequence can denote such a sequence (subsequence) as , that is, . This amounts to restricting the domain to a subset of natural numbers. If we denote this subset as where Ҿ . Another way of denoting a subsequence is , then we is a subsequence of natural numbers, that is, , that is, . It is usually required that is an increasing sequence. There is one important subsequence that we often use. Namely, define Sometimes, we shall denote such a sequence as . . Example 22: Let . The first terms are . Take every second term to produce a sequence that starts . We can write it as or as . Sequences are important since they are very often used in computer science. They are frequently used in sums and products that we discuss next. Consider a (sub)sequence ѷ and add all the elements to yield ѷ ѷ To avoids the dots we have a short hand notation for such sums, namely 14 In the above, we use different indices of summation , or since they do not matter. What matters is the lower bound and the upper bound of the index of summation, and the sequence itself. In a similar manner we can define the product notation. For the above case instead of writing ѷ we simply write Example 23: Here are some examples: Exercise 2D: Find 1. 2. . . We have to learn how to manipulate sums and products. Observe that ҷ . We obtained exactly the same In the above we change the index of summation from sum for some . In general, when we change index , we must change the lower summation index from to to, say to , the upper summation index from to , and must be replaced by . Example 24: This is the most sophisticated example in this module, however, it is important that the reader understands it. We consider a special sequence called the geometric progression. It is defined as follows: Fix and define for . This sequence begins with Let us now consider the sum of the first terms of such a sequence, that is, (2) 15 Can we find a simple formula for such a sum? Consider the following chain of implications ҷ ҷ ҷ ҷ where the second line follows from the change of the index summation factor in front of the sum, while in the last line we replaced , in the third line we by as defined in (2). Thus we prove that ҷ ҷ from which we find : as long as ҷ . Therefore, the complicated sum as in (2) has a very simple closed-form solution given above. An unconvinced reader may want to verify on some numerical examples that these two formulas give the same numerical value. 16 ...
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- Fall '08
- Set Theory, Natural number, Equivalence relation