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# Math 129 Week Two Notes Math 129 002

UNM

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This 3 page Class Notes was uploaded by StarvingArtistInCollege on Friday January 22, 2016. The Class Notes belongs to Math 129 002 at University of New Mexico taught by Dr. Karen K. Champine in Spring 2016. Since its upload, it has received 11 views. For similar materials see A Survey of Math in Mathematics (M) at University of New Mexico.

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Date Created: 01/22/16

Survey of Math: Week Two Sets Set: A list of elements separated by commas. Order of elements does not matter. Multiplicity does not matter o Ex: {a, a, a, b, c, c} is the same as {a, b, c} The elements are always contained in {} brackets. This is how you know if you are dealing with a set. Two sets are equal if they have the exact same elements in them. Ex: {a, b, c} = {c, b, a} Cardinality of sets: The number of distinct elements in that set. Ex: If set A= {a, 1, b, 2, c, 3}, then the cardinality of A is six, or │A│= 6, or n│A│= 6. Cardinality can be expressed as │x│= c or n│x│= c where x is the label for the set and c is the cardinality. The empty set: A set with a cardinality of zero. It is expressed as ø. A specific element in a set is expressed as Є. Ex: If A={1, 2, 3}, then 2Є│A│. A subset is expressed as C. Set A is a subset of Set B if all of the elements of A can be found in B. Ex: If A= {a, b} and B= {a, b, c, d}, then A C B. The empty set is always considered a subset. A proper subset is a subset which is not equal to the original set. If B= {a, b, c, d} and D={a, b, c, d}, then D C C, but it is not a proper subset because they are equal. If E={a, b, c}, then E is a proper subset of both B and D because E is not equal to them. The universal set is the set which contains all elements. Every possible set is a subset of the universal set. Well-Defined vs. Not Well-Defined Sets: A well-defined set is a set where we are exactly sure of what is in the set. For a set to be not well-defined, the set must contain things which are not certain. Ex: A set containing all the tall men in the world is not well-defined because the term “tall” is vague and can change from person to person. However, a set containing all men taller than 6 feet is well defined because the parameters are specific and clear. For two sets to be equivalent, they must have the same cardinality. Equivalent sets are expressed with ≈ Ex: If A={a, b, c} and B={1, 2, 3}, then A≈B Intersection: Expressed as A∩B It is the set of all elements that are in A and B o Ex: If A={a, b, c, d, e} and B={b, d, g, h}, then A∩B={b, d} If two sets do not have any elements in common, then their intersection is the empty set. o Ex: If A={a, b, c, d, e} and C={1, 2, 3}, then A∩C=ø Union: Expressed as AUB It is the set of all elements in A or B. o Ex: If If A={a, b, c, d, e} and B={b, d, g, h}, then AUB={a, b, c, d, e, g, h}. The intersection of a set is a subset of their union. So, A∩BCAUB If the intersection of two set is equal to the union of those two sets, then the two sets must be equal. Calculating Subsets: Generally, the number of subsets in Set S is the cardinality of S, or n(S), raised to the power of two. o Ex: If A={a, b, c, d, e}, then the cardinality of A is five, or n(A)=5. If we raise 5 to the power of 2, we get 32. This means that there are 32 possible subsets for set A. The power set is the set of all subsets in a set (confusing definition, right?) Ex: If set A={a, b,} then its power set is {ø, {a}, {b}, {a, b}}. Compliments: Expressed as A' (can also be seen as ~A, U/A or U-A) A compliment is anything in the universe that is not seen in a set. A not B: Expressed as A-B or A/B The part of a set that excludes the common elements of another set. Ex: If A= {a, b, c, d, e} and B= {b, c, e}, then A-B={a, d}. DIsjoint Sets: Two sets are disjoint if they have no common elements. In other words, when the intersect of two sets is the empty set.

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