# Section 8.0 (Preliminary) Sets, Set Operations & Cardinality

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1 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 1 A. Is 6 an element of A? It is not, so we write 6 A, 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. 2 Example 4: What is the solution set to the equation x + 2x 15 = 0? 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: 2 x + 2x 15 = 0 ( x + 5)( x 3) = 0 x = 5 or x = 3. Thus, the solution set to this equation is { 5,3}. 315

2 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, }. 316

3 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 be entirely objective and not open to opinion or subjectivity. Consider the following examples: Example 11: The set of rational numbers consists of all possible ratios of integers (positive and negative), including the integers themselves since integers can be made into a ratio by dividing by 1. However, it is rather difficult to list the rationals to establish a pattern (try it). Instead, it is considerably easier (and preferred) to represent the set using a descriptive rule for rationals. State the set of rationals using set-builder notation. a Discussion. The set is often presented as { x x = b ; a, b Integers, b 0}. This rule describes the appearance of all rationals beyond all doubt. Recall that dividing by zero is not a permissible arithmetic operation, hence the b 0 requirement. Example 12. Texas has 254 counties, much too large to simply list them (although not impossible). In such a case, set-builder notation is preferred. State the counties of Texas as a set. Discussion. The set can simply be written as { x x is a county in Texas }. The description is clear and unambiguous. Do you know all the counties in Texas? Probably not, but the description is still well-defined; with the help of an atlas, you can determine if a candidate county x is a county in Texas, or if it is not. The description offers no ambiguity. 317

4 Example 13. Consider the set { x x is a good motion picture }. Discuss its merits. Discussion. The description is ambiguous and subjective, hence is not a well-defined set. Is Slap Shot an element of this set? Most would say absolutely yes, but there may be a few people who would disagree. The elements of this set could never be determined exactly. Example 14. You are given the set G = { x x is a positive integer that is an integer multiple of 3, but not of 4 }. List some elements of G. Discussion. The description is unambiguous and well-defined. You can easily check to see if a candidate x is a member of G, or not. With a little work, we see that 3 is in G, while 4 and 5 are not. 6 is in G, 7 and 8 are not. 9 is in G, but 10, 11, 12, 13, and 14 are not (Do you see why 12 is not?). In fact, this description lends itself to a pattern: G = {3, 6, 9, 15, 18, 21, 27, 30, 33, 39, 42, 45, }. We leave it up to your to decide which form is preferred in this example. The Empty Set and the Universal Set We now look at two special sets that are vital to a complete understanding of the nature and arithmetic of sets: the empty (null) set and the universal set. A set that contains no elements is called the empty set or the null set, and is written. On the other extreme, the universal set (or universe) is the set of all elements, within the context of a given problem. The common symbol for universe is U. In the vast majority of cases, the universe is either explicitly stated, or is plainly obvious. Consider the following examples: Example 15: What is the solution set for the equation x = x + 1? Discussion. When attempting to solve for the variable, it cancels itself out of the equation, leaving an absurd 0 = 1 statement, which is false for all values of x. Therefore, the solution set to this problem is the empty set for there is no value x that will solve this equation (make the equation true). Example 16: State some common examples of universal sets. Discussion. In Real number algebra, the set of numbers from which we work is the Real Numbers. This is the (presumptive) universe for virtually all algebraic equations. Occasionally we might use a different universe: for example, when working with negative quantities under the radical, we may allow the use of the Complex Number Field as a universe. As a child, your universe of numbers was likely the positive counting Whole integer values. If we are spelling words, the universe is the set of letters in the alphabet. Words in this book come from the universe of words in the English language. Can you think of other universes? 318

5 Set Arithmetic The next natural step in handling sets is to relate one set to one another, and to combine sets together in useful, meaningful ways. Just as numbers can be compared via the relations greater than, equal and less than, and combined using arithmetic operations such as addition and multiplication, sets can be manipulated in similar ways. First, a useful diagramming tool called a Venn Diagram is introduced. Venn Diagrams, Equality and Subset A useful method to visualize sets and how they relate with one another is the Venn Diagram. Venn Diagrams are sketches of circles and other simple regions representing sets. The usual protocol is to draw a rectangle to represent the universe (place the U on its outside). Sets A, B, etc, are represented by overlapping circles so as to allow for potential common elements. The typical Venn diagrams for 1, 2 and 3 sets are given in the following figures: Figure. When we have a single set A and a universal set U, we can illustrate their relative appearance by exhibiting set A as a circle inside the larger universe U. Figure. If A and B are two sets within the universe U, then we arrange the sets in the above manner. Four regions result from this arrangement. Figure. If A, B and C are three sets within the universe U, we arrange the sets as in the above manner. Eight regions result from this arrangement. 319

6 We use Venn diagrams to help visually display the various set arithmetic operations. They also serve as useful tools for counting problems, and as we shall see in the upcoming sections, many probability problems. We next consider the relation equality as it relates to sets: Two sets are equal if they contain the same elements, disregarding any repeats and the order in which the elements are listed. The equality relation is reflexive ( A = B implies B = A and vice-versa) and transitive ( A = B and B = C imply A = C ). Example 17: Consider the following sets. Are they equal? A = {1, 2, 3, 4, 5} B = {5, 4, 3, 2, 1} C = {1, 1, 2, 3, 4, 5} D = {1, 1, 1, 3, 3, 4, 2, 2, 5, 4, 5, 3 } Discussion. All four sets are equal, as they all contain the elements 1, 2, 3, 4, and 5. Reordering the elements and repeating the elements makes no difference. We write A = B, etc. Set A is a subset of set B if all elements of A are also elements of B. The symbol for subset is, and we write A B. The formal definition of subset is: A B = {x if x A, then x B } The empty set is a subset of all sets by definition. Informally, if A B, we often view set A as a smaller portion of set B, or that set A is contained inside set B. Similarly, B contains A, and B is called a superset of A. Figure. In the left diagram, A is a subset of B, and B is a superset of A.. Both are subsets of U; in fact, A is a subset of B which is a subset of U: A B U. In the right diagram, A is not a subset of B, nor is B a superset of A. However, independently, both are subsets of U: A U is true, as is B U. If A is a subset of B, then a new set called B A ( B minus A ) can be formed. B A is the set of elements in B but not in A. Furthermore, B A is also a subset of B. 320

7 The notions subset and superset are relations, in that the statement A B is either true or false. Example 18: Let A = {1,2,3} and B = {1,2,3,4,5}. Is A B? Discussion. Since all elements in A are also elements in B, we can conclude that A is a subset of B, and write A B. Incidentally, B A = {4,5}, and this is also a subset of B: ( B A) B. Figure. A visual representation of the example. Elements 1, 2 and 3 are in A as well as in B (do you see why?). A is contained in B. A is a subset of B. A set A is a proper subset of B if all elements of A are in B, while B may contain some elements not in A. In other words, while A B is true, A = B is not. The symbol for proper (strict) subset is, and is written A B. Any set A is an improper subset of itself. The statement the statement A A is false. A B is true, and so is A = A. However, The set of people who live in New York City (NYC) is a proper subset of people who live in New York State (NYS). The statement NYC NYS is true. So is the statement NYC NYS. However, the statement NYC = NYS is false. The set of people who live in New York City is an improper subset of itself; the statements NYC NYC and NYC = NYC are both true, but NYC NYC is false. Example 19: Let A = {1,2,3} and B = {2,3,4,5,6}. Is A B? Discussion. Not all elements in A are elements of B, specifically, the element 1 in A is not an element of B. A is not a subset of B. Figure. In this example, the element 1 is a member of set A but not of set B. Since not all elements of A are members of B, then A cannot be considered a subset of B. 321

8 Example 20: Consider the following sets. Which sets are subsets of one another? Which are improper subsets of one another? Which are proper subsets of one another? A = {1,2,3} B = {1,2,3,4,5} C = {1,2,4,5,6} D = {1,2,3,4,5,6,7} Discussion. As we have already seen, A B. It is also true that A D, B D and C D. These are all examples of proper subsets, in which case it is also correct to write A D, B D and C D. A is an improper subset of A, as is B with B, C with C and D with D. Note that A is not a subset of C (do you see why?). We could write this as A C. The same is true for sets B and C (again, do you see why?). Example 21: Given U = {a, b, c}, write out all subsets of U. Which are proper? Which are improper? Discussion. By definition, the empty set is a (proper) subset of U. The sets {a}, {b}, {c}, {a,b}, {a,c}, and {b,c} are all proper subsets of U. The set {a, b, c} is an improper subset of U. Beware! There is often confusion between the notions of element of and subset of. The notion of subset refers only to sets, not individual elements. However, we can create a set of just one element (called singletons), thus allowing us to use the subset concept if need be. For example, if set A = {1,2,3,4,5}, then we could say that 2 is an element of A : 2 A, and we could also say that the set containing 2 is a subset of A : { 2} A, where we are treating the element 2 as a subset by enclosing it in set brackets. It is not permissible to mix the terminology: we can not say { 2} A, nor can we say 2 A, for example. Fact. If A is a subset of B, then so is B A. Since B is a subset of itself, this would imply that B B is also a subset of B. The only sensible meaning to B B is the empty set (i.e. B B is all the elements of B removed from B, leaving nothing). Hence, the empty set is a subset of B. Complement, Union and Intersection. As mentioned previously, equality and subset are relations between two sets: statements such as A = B and A B can only be answered as true or false. On the other hand, complement, union and intersection are operations, since we can combine two sets together and form a new third set as a result. The complement of a set A is the set of all elements not in A, relative to some universal set U. The symbol for complement is to append a small prime, viz. A. Formally, A = { x x A ( x U )} 322

9 The presence of the universal set U is vital. Without it, there is virtually no limitations on what the appearance of A could be. It should be intuitively obvious that A is a subset of U, and that A is also a subset of U. A could also be written U A. Figure. In the left diagram, set A is shaded, while in the right diagram, the complement of A is shaded. Example 22: Let U = {1,2,3,4,5} and A = {1,2,4}. Find A. Discussion. A = {3,5}. Compose A by choosing those elements in U that are not in A. Note that sets A and A are proper subsets of U. Example 23: Let U = {1,2,3,4,5} and A = {1,2,3,4,5}. Find A. Discussion. Since there are no elements in U that are not in A, then A =. Note that A is an improper subset of U. Since a set A and its complement A are subsets of the universal set U, we must have that the empty set is also a subset of U. This lends credence that the empty set is a subset of all sets. Example 24: Let U = {1,2,3,4,5} and let A = {1,2,3}. Find A. Discussion. The expression A is read as the complement of the complement of A. Since A = {4,5}, then the complement of A must be {1,2,3}, which is our original set A. This example illustrates the fact A = A for all sets A. The union of two sets A and B is the set composed of those elements in A or in B, possibly both. The symbol for union is. Formally, A B = { x x A or x B} Informally, the union of two sets is formed by combining the two sets together into one large set. Figure. The Union of two sets A and B is represented by the shaded region. 323

10 Fact. Sets A and B are both subsets of the union to both sets A and B. A B. Equivalently, the union A B is a superset Example 25: Let A = {1, 2, 3} and B = {2, 4, 6, 7}. Form the union A B. Discussion. The union A B = {1, 2, 3, 4, 6, 7}. Note that 2 is an element of both sets simultaneously. Even so, it meets the requirements for membership in the union A B. Figure. The union of sets A and B of the given example. The intersection of two sets A and B is the set composed by those elements in A and B simultaneously. The symbol for intersection is. Formally, A B = { x x A and x B} Informally, the intersection can be thought of as the overlap between the two sets A and B. Figure. The intersection of sets A and B is represented by the shaded region. Example 26: Let A = {1, 2, 3} and B = {2, 4, 6, 7}. Form the intersection A B. Discussion. The intersection A B = {2}. The element 2 is the only element common to both A and B simultaneously; we sometimes say that the element 2 is common to both sets A and B. Figure. The intersection of the two sets A and B of the given example. 324

11 If two sets have no common elements, then their intersection is empty, and the two sets are considered mutually exclusive. If three or more sets have no elements in common, then each pair of nonintersecting sets are called pairwise mutually exclusive. Example 27: Let A = {1, 2, 3} and B = {4, 5, 6}. What is Discussion. Since A and B have no elements in common, mutually exclusive. A B? A B = and sets A and B are considered Fact. The intersection Figure. The shaded region contains no elements; the intersection is empty. A B is a subset of both sets A and B separately. The statements ( A B) A ( A B) B are both true for all sets A and B. Furthermore, the intersection of sets A and B will always be a subset of the union of two sets A and B. The statement is always true for all sets A and B. ( A B) ( A B) Fact. The union of a set A with its complement A is always the universe. A A = U is always true for all sets A. In fact, if A = U, then A =, and since both a set and its complement are subsets of the universe, this implies that the empty set is also a subset of the universe, as we had previously asserted. Fact. The intersection of a set A with its complement A is always empty. A A = is always true for all sets A. This should be very intuitive: an element is either a member of a set or a member of its complement, but never both, and never neither. 325

12 Compound Set Operations We can combine the basic set operations union, intersection and complement to form compound set operations. Any expression of sets involving more than one set operation is considered to be a compound expression. Example 28: Let U = {1,2,3,4,5,6,7,8}, A = {1,2,4,6,8} and B = {2,3,4,5,6}. Find A B. Discussion. The expression A B features complement and intersection, and is considered to be compound. We work the sets in the following order: we find B first, then intersect it with set A: B = {1,7,8} A B = {1,2,4,6,8} {1,7,8} = {1,8} The final result is A B = {1,8 }. You may find it helpful to actually write out each set at each step. Figure. This Venn diagram shows the relative relationship of the elements. The expression A B can be read as those elements that are inside set A and outside set B In this example, only elements 1 and 8 meet this requirement. The above example and Venn diagram suggest a way to verbally describe the possible relationships that exist between two sets A and B. Note that in the standard representation, there are four regions that result. They are: The intersection of A and B: those elements inside A and inside B : A B. The intersection of A and B : those elements inside A and outside B : A B The intersection of A and B: those elements outside A and inside B : A B The intersection of A and B : those elements outside A and outside B : A B A Venn Diagram illustrating these regions is on the next page: 326

13 Figure. The four descriptive relationships of two sets A and B, along with its associated compound set expression. Example 29: Let U = {1,2,3,4,5,6,7,8}, A = {2,3,4,6,7}, and B = {4,5,6,8}. Find the following sets ( A B) and A B Discussion. For set ( A B), we work inside the parentheses first to find A B, then find the complement of that set: A B = { 2,3,4,5,6,7,8} ( A B) = {1}. For the expression A B, we find the individual complement sets first: A = { 1,5,8}, B = {1,2,3,7} Then we intersect the two sets: { 1,5,8} {1,2,3,7} = {1} In both cases we arrived at the same result, the set {1}. This example illustrates an identity known as DeMorgan s Laws: DeMorgan s Laws Complement of the Union: ( A B) = A B Complement of the Intersection: ( A B) = A B DeMorgan s Laws offer a way to distribute the complement across union or intersection. Essentially, distribute the complement to the individual sets, and switch the union with an intersection (or vice-versa). These laws help reduce very cumbersome and confusing expressions into much simpler ones. 327

14 Example 30: Use DeMorgan s Laws to reduce ( A B). Discussion. We distribute the complement and switch the intersection for a union: ( A B) = A B = A B Note that in the second line, we get (temporarily) the complement of the complement of A, which as we have already seen, reduces to A. Cardinality An immediate application of set arithmetic and Venn diagrams involves counting; specifically, an effective way to handle quantities and to perform meaningful addition between (possibly overlapping) sets. The cardinality of a set is the number of elements contained in the set. If A is a set, the cardinality of set A is written n (A). The empty set has cardinality zero: n ( ) = 0. Example 31: State the cardinality of the following sets. A = {2,3,4,5,12} B = {a,e,i,o,u} C = {dog, cat, giraffe, goat, duck, rock} D = {0} E = Discussion. Set A contains 5 elements, hence n ( A) = 5. In the same manner, n ( B) = 5 and n ( C) = 6. In regards to sets D and E, we see an important relation and distinction between the concepts of zero and empty set. Zero 0 is the numerical quantity representing nothing, while the empty set is the set containing no elements. Although they have analogous meanings, the two concepts (and symbols) 0 and are not the same thing. In this example, Set D is the set that contains the number 0 (it is therefore not empty) and so n ( D) = 1. On the other hand, E contains no elements; it is the empty set, so n ( E) = n( ) = 0. When two sets have the same cardinality, the two sets are equivalent to one another. In the above example, sets A and B are equivalent to one another, but not equal. The symbol for equivalence is. Thus, the statement A B is true, while A = B is false. 328

15 Cardinality of the Union of Two Sets Consider two sets A and B, and their union A B. We want to count the number of elements in the union; that is, to calculate c( A B). It seems sensible to count the elements in A, and then count the elements in B, and add the two quantities together. However, there is the possibility that some elements have been counted twice, specifically, those elements that lie in both sets A and B simultaneously the intersection A B. In order to balance this over-count, we subtract the number of elements in the intersection. The result is a useful formula called the cardinal of the union: Cardinal of the Union: n( A B) = n( A) + n( B) n( A B) The cardinal of the union formula can also be used to find the quantity in an intersection as long as the other values are known, just by simple algebra. Furthermore, the formula works quite well in conjunction with Venn Diagrams, as we shall see in upcoming examples. Example 32: Let A = {1, 2, 3} and B = {2, 4, 6, 7}. Calculate the number of elements in the union B A. Discussion. Note that n ( A) = 3 and n ( B) = 4. Also note (from example 26) the intersection A B = {2}, so n ( A B) = 1. The formula then gives: n( A B) = n( A) + n( B) n( A B) n( A B) = n( A B) = 6 Hence, there are 6 elements in the union. By observation, we see that this is true: 6, 7}, which contains 6 elements. A B = {1, 2, 3, 4, The following survey example illustrates many of the concepts of sets, including Venn Diagrams, union, intersection and cardinality: Example 33: 50 college students are surveyed and the following results are obtained: 32 are currently taking a math course, 27 are currently taking a history course, and 18 are taking one of each (both). How many students are taking math or history? Discussion. Let set M consist of the students taking a math course and set H consist of the students taking a history course. Therefore, n ( M ) = 32, n ( H ) = 27 and n ( M H ) = 18. As a result, we obtain: n( M H ) = n( M ) + n( H ) n( M H ) n( M H ) = n( M H ) =

17 Figure. Process for filling in a three-way Venn diagram, example 34. Now, all of the question posed in the example can be answered, just by careful reading of the Venn Diagram. Be sure to note the significance of the words or (for union) and and (for intersection). Question (a) was How many people had been to Atlanta or Boston? This is the union A B. Hence, n ( A B) = = 27. Question (b) was How many people had been to Boston and Chicago but not Atlanta? This is the intersection B C A. This is just a single region containing 2 elements: n ( B C A ) = 2. Question (c) was How many people had been two exactly two of the cities? These are the regions denoted by the intersections A B, A C and B C but excluding the three-way intersection A B C. Identify the areas that meet these conditions, and sum their elements: = 9. Question (d) was How many people had not been to Boston or Chicago? This is the complement of the union of B with C, or in symbols, ( B C). Sum the appropriate regions to get n (( B C) ) = 11. Question (e) was How many people have been to neither of the cities? This is the complement of the unions of A, B and C. Hence, n (( A B C) ) = 7. The Venn Diagrams for each of the five questions above are given below, with the appropriate regions shaded in: Figure. Shaded regions corresponding to the questions of example

18 Summary A set is a collection of items, each item is an element of the set. Sets can be written using roster notation or set-builder notation. A set must be well-defined. No ambiguity is allowed. The empty set contains no elements and is written. The universal set contains all elements in context to the scenario. Two sets are equal if they contain the same elements, repeats and ordering ignored. Set A is a subset B if: A B = {x if x A, then x B } The complement of a set A is: A = { x x A and x U } ( x is not in A ) The union of two sets A and B is: A B = { x x A or x B} The intersection of two sets A and B is: A B = { x x A and x B} Two sets are mutually exclusive if their intersection is empty. The cardinality of a set is the number of elements in the set. Two sets are equivalent if their cardinalities are the same. 332

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