Section 8.0 (Preliminary) – Sets, Set Operations & Cardinality 315 Section 8.0 (Preliminary) Sets, Set Operations & Cardinality A set is simply the grouping together of objects. For example, we count using the set of whole numbers 1, 2, 3, 4, and so forth. We spell words using a set of symbols (letters) known as the alphabet. Geography students might be able to name the set of fifty states of the United States. These are a few of many such examples of the grouping together of objects as a set in some useful manner. Sets and Elements A set is a grouping of objects. Each object in the set is called an element of the set. The symbol for element is ∈. Sets are usually identified by a capital letter, and the elements of the set are always enclosed within curved brackets { }. Example 1: Consider the set A = {1, 2, 3, 4, 5}. Discussion: The set A is the set of whole numbers 1, 2, 3, 4, 5. Conversely, the numbers 1, 2, 3, 4, and 5 are all elements of set A. Consider the element 1 for a moment. In English, we would say that “1 is an element of A”. In set notation, we write A∈1. Is 6 an element of A? It is not, so we write A∉6, and state that “6 is not an element of A”. Example 2: The alphabet is a set of symbols: {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}. Discussion: Each symbol (letter) is an element of the alphabet. For example, “k ∈ alphabet” is a true statement. Example 3: Write the set of even positive integers less than or equal to 20. Discussion: This set is {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}. The order in which the elements are written does not matter. Example 4: What is the solution set to the equation 01522=−+ xx? Discussion. The solutions (if they exist) of an algebraic equation are listed as a set, called the solution set. Using a bit of algebra, we factor the polynomial and set each factor to zero, and solve: 50)3)(5(01522−=→=−+→=−+ xxxxx or 3=x. Thus, the solution set to this equation is }3,5{−. Section 8.0 (Preliminary) – Sets, Set Operations & Cardinality 316 Example 5. Let S = the set of states of the United States of America. Is Manitoba ∈ S? Discussion. The set S of the states of the United States of America is described descriptively as opposed to having each of the 50 states listed. Regardless, this set is very clear as to what it contains (and does not contain). For this set, Manitoba ∉ S. Roster and Set-Builder Notation; Well-Defined Sets. The last example showed that a set can be described in descriptive terms as opposed to rote listing of its elements, which may not be convenient to do. This leads us to two common ways to represent a set: roster notation and set-builder notation. • In roster notation, the elements of the set are written individually, separated by commas. Ellipses (…) may be used to denote the continuation of a pattern. • In set-builder notation, the elements of the set are described according to some objective rule. If the set is small, or follows some very distinct and obvious pattern, then roster notation is usually adequate. If the set is large and does not follow an obvious pattern, then set-builder notation is the better choice. Consider the following examples of roster notation. Discuss the merits of each example. Example 6: A is the set of positive integers less than or equal to five. Write set A. Discussion: This set is small enough to simply list its elements. Roster notation is adequate for presenting this set. Hence, A = {1, 2, 3, 4, 5}. There is no significance to the order in which the elements are listed. Set A could also be written {5, 4, 3, 2, 1}, for example. Example 7: B is the set of positive integers less than or equal to 100. Write set B. Discussion: Clearly we are intending to present the entire set of positive integers from 1 to 100 inclusive. Due to its size (100 elements), we establish a pattern, let the ellipses (…) signify the continuation of the pattern, and exhibit the ‘end’ number so that it is very clear to the reader the exact composition of this set. Hence, B = {1, 2, 3, 4, 5, … , 100}. Note that enough elements should be written so that the pattern is obvious. See example 9, next page. Example 8: Set C is the set of positive even integers. Write set C. Discussion. Similar to set B, we establish a pattern for set C and follow it with the ellipses to signify the continuation of the pattern. However, there is no ‘end’ number; this set apparently represents all positive even integers, as implied by the pattern. Hence, C = {2, 4, 6, 8, 10, …}. Section 8.0 (Preliminary) – Sets, Set Operations & Cardinality 317 Example 9: Consider the set D = {1, …, 99}. Describe this set in words. Discussion. There is more than one way to interpret “1, …, 99”; for example, is this the set of integers from 1 through 99, or, possibly, the set of odd integers from 1 to 99? There are many reasonable ways to interpret set D, which unfortunately makes it unclear as to what exactly D is. Avoid ambiguity by describing the elements of the set in clearest possible terms. If the set follows a pattern, write out at least four or five elements to strongly establish the pattern. If the pattern is not easily described by writing out a few of the elements, use set-builder notation to describe exactly what you intend for the set to contain. Example 10: Consider the set J = {2, 3, 5, … }. Discuss its merits, or lack thereof. Discussion: The pattern here still leaves room for ambiguity. Is it the set of prime integers or the set of Fibonacci numbers greater than 1? We are not sure, and as a result, it is not obvious just exactly what is and is not an element of J. The last two examples illustrate a possible problem when presenting a set: it may not be practical to list all elements or perhaps the elements do not follow an obvious pattern. In such a case it is wise to forego roster notation and instead describe the set using set-builder notation. Set-builder notation follows the standard format: { x | ‘some property of x’ } The set is then translated as “the set of x, such that x meets the stated property.” The property must be well defined. The property should
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