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UMD MATH 111 - Sets and Set Operations

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Math 111, section 6.1 Sets and Set Operations notes by Tim Pilachowski Our first topic is one which will (hopefully) clarify the ideas underlying probability. Definition: A set is The individual objects in a set are called Every set has three properties. 1. 2. 3. Sets are usually represented by capital letters, such as A, B, C, etc. Notationally, a set is indicated using braces (squiggly brackets). The elements of a set can be defined as a descriptive sentence, list, or equation. G = the set containing the letters “x”, “ö”, “A”, and the integers “0”, “9”, “12” = {0, x, ö, 9, 12, A} H = the set of colors of lights in a standard traffic signal = I = the set of “solutions to the equation x2 = 4” = {x| x2 = 4} read “the set of elements x such that 42=x”. This version is called set-builder notation. So, as a list, I = J = the set of positive even numbers = The symbol ∈ means “is an element of”. Examples A: Let G, H, I, and J be as defined above. The symbol ∉ means “is not an element of”: Examples A (continued): Let G, H, I, and J be as defined above.Two sets A and B are called equal, i.e. A = B, when they have exactly the same elements. When a set is defined by listing its elements the list may be in any order. Examples B: Let H = {red, yellow, green} If a set has no members, it is called the empty set or the null set, and is denoted either by empty braces, { }, or by the symbol ∅. Important note: One set A is a subset of another set B if every element found in set A is also in set B. Another way to say this is that there is nothing found in A which is not also found in B. Examples C: Let I = {x | x2 = 4} = {– 2, 2}, J = {2, 4, 6, 8, …}, and K = {4, 8, 12}. Note the difference between a “subset” and a “proper subset”, in both concept and notation. Theorems about subsets: Examples C revisited: Let I = {x | x2 = 4} = {– 2, 2}, J = {2, 4, 6, 8, …}, and K = {4, 8, 12}. List all of the subsets of I. List all of the subsets of K. (Note that set J would have an infinite number of subsets.)We’ll be considering three fundamental set operations. From two given sets A and B we can make a new set that consists of all the elements of A and all the elements of B. This new set is called the union of A and B and is represented by the symbol BA∪. (The union symbol is not the letter U.) Example D: Let Q = {a, b, c, d} and let R = {c, d, e}. The union of two sets is defined in symbols as follows: {}BxAxxBA ∈∈=∪ or | . Note that this is a non-exclusive use of the word “or”: the elements can be in A, or in B, or possibly in both A and B. Other notes: From two given sets A and B we can make a new set that consists of all the elements that belong to both A and B at the same time. This new set is called the intersection of A and B and is represented by the symbol BA∩. Example D (continued): Let Q = {a, b, c, d} and let R = {c, d, e}. The intersection of two sets is defined in symbols as follows: {}BxAxxBA ∈∈=∩ and | . Other notes: Two sets whose intersection is empty are called disjoint, i.e. two sets A and B are disjoint if and only if ∅=∩BA . Before addressing the operation of complement, it is necessary to define a universal set, containing all the individual objects under consideration. For example, if the sets being studied consist of men, women, boys, and girls in a population, then the universal set is everyone in the population. In a primary school mathematics classroom, the universal set contains only positive rational numbers. In a typical algebra classroom, the universal set contains all real numbers, positive and negative, rational and irrational. The letter U (not the union symbol ∪) is used to denote the universal set for a given situation. The complement of a set A is the set of all elements in the universal set that are not members of A, and is represented by the symbol cA. It is defined in symbols as follows: {}AxUxxAc∉∈= and | . Example E. Let U = the set of positive whole numbers = {1, 2, 3, 4, … } and J = the set of positive even numbers = {2, 4, 6, 8, …}.Complement literally means “that which completes”, and if you combine a set with its complement, you get everything, i.e. the universe. Other notes: Venn diagrams provide a visual means of considering sets, even when the particular elements may not be known. In a Venn diagram a rectangle represents the universe set under consideration and circles within the rectangle represent sets within the universe. The operations above would be diagrammed as follows: BA∪ BA∩ ∅=∩BA (one way) ∅=∩BA (another way) cA BA⊂ Venn diagrams can also be used with three (or more) sets. Example F: Draw a Venn diagram to illustrate first ()CBA ∩∪ then ()cCBA ∩∪ . On your own: Use a Venn diagram to show that ()()CBACBA ∩∪≠∩∪ . ()CBA ∩∪ ()cCBA ∩∪ ()CBA ∩∪Your text introduces the distributive laws for unions and intersections ()()()CABACBA ∪∩∪=∩∪ and ()()()CABACBA ∩∪∩=∪∩ as well as De Morgan’s Laws ()cccBABA ∩=∪ and ()cccBABA ∪=∩ but none of these will be useful enough in this course for you to worry about memorizing them. Proving them for yourself, using Venn diagrams, can be a good exercise. Examples G: Consider the universe U = {a, b, c, d, e, f, g, h, i, j, k} and sets M = {a, b, c, d}, N = {b, c, d, e, f, g}, and P = {g, h, i}. =cM =∪NM =∩NM =∪PM =∩PM =∪cPN =∩cPN ()=∪∩cPNM ()=∩∪cPNM Now let T = {c, e, b, d}. What can we say about T in relationship to M, N, and NM∩ ? Example H: Let U = the members of a freshman class at UMCP, A = the set of students who have academic scholarships, B = the set of students who are athletes, and C = the set of students who live on campus. Describe each set below [parts a) through e)] in words. a) cA b) CB∩ c) cBA∩ d) BA∪ e) cCB ∪Write the set that represents (symbolically) each statement f) through k) below, then also draw a Venn diagram to illustrate each set. f) the set of students who do not live on campus g) the set of athletes who have academic scholarships A B C A B C h) the set of athletes who don’t live on campus i) the set of students who have academic scholarships …


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