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Unformatted text preview: SHENS CLASS NOTES191 1 Chapter Two The Language of Mathematics In this chapter, we discuss sets, functions, and sequences. These are very fundamental notions or language to model mathematical problems as well as tools for solving mathematical problems. 2.1 Sets Definition 2.1 A set is an unordered collection of objects. The objects in a set are also called elements or members. Example 1 A = {1, 2, 3, 4} is a set that contains four numbers. Example 2 V= { a , e , i , o , u } is the set of all vowels in English alphabet. One way to define a set is to list all members of the set as shown in Example 1 or 2. SHENS CLASS NOTES191 2 Another way is to use the set builder notation. For example, R = { x  x is a real number}, B = { x  x is a positive, even integer}. The following are some commonly used sets: N = {0, 1, 2, 3, }, the set of natural numbers. Z = {, 1, 0, 1, 2, }, the set of integers. Z + = {1, 2, 3, }, the set of positive integers. Q = { p / q  p Z , q Z , q 0}, the set of rational numbers. R , the set of real numbers. { } or , the empty set. The size of a set S is called its cardinality , denoted by S. If S is a finite set, consisting of n elements, than  S  = n . For example, {1, 2, 3, 4} = 4. Definition 2.2 Two sets are equal if and only if they have the same elements. SHENS CLASS NOTES191 3 Example 3 (2.1.1) Let A = { x  x 2 + x6 = 0} and B = {2, 3}, prove A = B. Proof. If x A, then x satisfies x 2 + x 6 = 0, which means that x is a root of the equation. Notice that x 2 + x 6 = ( x  2)( x + 3). This equation has two roots, 2 and 3. Therefore, x must be either 2 or 3. Obviously, any x that is in set A must be in set B also. Conversely, any element of set B must be either 2 or 3 which are the roots of x 2 + x 6 = 0. It must belong to set A also. Therefore, A = B. Definition 2.3 Suppose that X and Y are sets. If every element of X is an element of Y, we say that X is a subset of Y , denoted by X Y . Y is also called a super set of X . Example 4 (2.1.2) C = {1, 2}, A = {1, 2, 3, 4}, C A. We see that A B if and only if x ( x A x B ). Definition 2.4 Suppose that X is a subset of Y but X Y , then SHENS CLASS NOTES191 4 we say that X is a proper subset of Y . denoted by X Y . Definition 2.5 The power set of a set X is the set of all subsets of the set X , denoted by P ( X ). Example 5 (2.1.5) If A = { a , b , c }, then P ( A ) = { , { a }, { b }, { c }, { a , b }, { b , c }, { a , c }, { a , b , c }}. Theorem 1 (2.1.6) If  X  = n , then  P ( X ) = 2 n . Proof. Let P ( n ): If  X  = n , then  P ( X ) = 2 n . We prove the claim P ( n ) by induction on n ....
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 Fall '06
 Shen

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