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# MATH-M 018 Exam 1 Study Guide MATH-M 18

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This 4 page Study Guide was uploaded by Abigail Dolbeer on Wednesday September 7, 2016. The Study Guide belongs to MATH-M 18 at Indiana University taught by Linda McKinley in Summer 2016. Since its upload, it has received 10 views. For similar materials see Basic Algebra For Finite Math in Mathematics at Indiana University.

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Date Created: 09/07/16

MATH-M 018 Exam 1 Study Guide Ch 1: Sets Sets: a collection of objects (elements) Sets are generally noted by a capital letter The order in which the elements appear in the set is not important Notation of sets: list each element, separated by a comma, surrounded by braces Example: {1, 3, 5, 7} or {1, 3, 5, …} Finite set Infinite set ∈ : is an element of ∉ : is not an element of A= {1, 4, 8, 10} so 4 ∈ set A and 5 ∉ set A Universal Set: a set that contains everything (that is relevant to the question) Equality: two sets are equal if they have precisely the same numbers Set A=B were A is the set whose numbers Are the first four positive whole numbers B= {1, 2, 3, 4} Empty Set: a set with no elements; ∅ symbolizes empty set Subset: when a set is defined, a subset can be formed with pieces of that set Ex: A is a subset of B where A= {1,3,4} and B= {1,4,3,2} ⊂ is a subset of ⊄ is not a subset Set Builder Notation: describes a set by saying what properties the set has {x | x > 0} = the set of all xs such that x is greater than 0 All x X is greater than 0 The set of Such that Complements: complements of a set contain every element not in that set The complement of S is denoted by S’ (pronounced S prime) S’ contains every element not in S Set Operations: Intersection: The intersection A ∩ B is the set that contains elements of A that also belong to B Union: the set of all distinct elements in the collection Venn Diagrams: a way of picturing an expression involving one or more sets Steps: 1. Draw a rectangle which represents the universal set. 2. Inside the rectangle, circles are used to represent various sets in the universe. 3. Circles are always overlapping to allow for the possibility that the sets may have elements in common. • Note that emptiness is a possibility Using Venn Diagrams in Problem Solving: 1. Define the steps 2. Enter the given information into a venn diagram Ch. 2 Probability and Counting Experiments and events: An experiment is any situation whose outcome is uncertain and can be thought of as an experiment. Experiments consist of finitely many outcomes. Probability theory is the field of mathematics that was developed to answer probability questions. Sample space of an experiment is the set of all outcomes of the experiment. A sample space is often denoted “S”. Complementary events: Probability: likelihood of something happening If the sample space for an experiment consists of n equally likely outcomes, then the probability we assign to each of the outcomes is 1/n. Suppose that E is an event in a sample space with equally likely outcomes. Then, the probability that event E will occur, denoted Pr(E), is: Pr(E)= n(E)/n(S) The relationship between the number of outcomes in an event & the number of outcomes in its complement is given by: N(E) + N(E’) = n(S) Relationship between the probabilities of an event and its complement: N(E)/n(S) + n(E’)/n(S) = n(S)/n(S) or Pr(E) + Pr(E’) = 1 The Multiplication Principle: states that if a multi-stage experiment has n1, outcomes at stage 1, n2 outcomes at stage 2 (regardless of the result of stage 1), n3 outcomes at stage 3 (regardless of the results of the first two stages), and so on until nk outcomes at stage k (regardless of the results of the previous stages) then the experiment has n1 x n2 x n3 x… x nk outcomes. • One can think of the multiplication principle as “filling slots”. Each slot represents a stage of the experiment, and the number you place in the slot is the number of outcomes at that stage. • Suppose that E is an event in a sample space with equally likely outcomes. Then the probability that event E will occur, denoted Pr(E) is: Pr(E)= n(e)/n(s) Permutations: an ordering of some set of elements If n is a positive whole number, the notation n! is read “n factorial” and stands for the product: n x (n-1) x (n-2) x (n-3) . . . 3 x 2x 1 If n is a positive whole number, then there are n! ways to order n different things Combinations: Combinations are different than permutations because combinations cannot be counted directly by “filling slots” because the multiplication principles always counts all possible orders Permutations: ordered lists, ordering, arrangement Combinations: selection, collection, (sub)sets The formula for calculating C(n,r) 1. The number of ways to order r things is r! 2. The number of ways to select and arrange r things from n is: P(n,r)= n!/(n-1)! The number of ways to select and order r things from n is the number of ways to select r things from n times is the number of ways to order them: P(n,r) = C(n,r) x r! The number of ways to select a set of r things from n is called the number of combinations of n things taken r at a time and is denoted C(n , r) where: C(n , r) = n!/r!(n-r)!

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