Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Sets and Counting6 Sets and Set Operations The Number of Elements in a Finite Set The Multiplication Principle Permutations and Combinations

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Set: A set is a collection of objects/elements. Ex. A = {w, a, r, d} Sets are often named with capital letters. Order of elements doesn’t matter, no duplicates. Set-builder notation: rule describes the definite property (properties) an object x must satisfy to be part of the set. Ex. B = {x | x is an even integer} Read: “x such that x is an even integer” Notation: w is an element of set A is written Section 6.1 Sets and Set Operations

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Set Equality: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. Subset: If every element of a set A is also an element of a set B then A is a subset of B, written Ex. A = {w, a, r, d}; B = {d, r, a, w} Ex. A = {r, d}; B = {r, a, w, d, e, t} Every element in A is also in B Every element in A is in B and every element in B is in A. Set Terminology and Notation

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Empty Set: The set that contains no elements is called the empty set and is denoted Universal Set: The set of all elements of interest in a particular discussion is called the universal set and is denoted U. Note: The empty set is a subset of every set Set Terminology and Notation Cont.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Set Union: Let A and B be sets. The union of A and B, written is the set of all elements that belong to either A or B. Set Intersection: Let A and B be sets. The intersection of A and B, written is the set of all elements that are common to A and B. Set Operations

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Given the sets: Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Venn Diagrams U AB – visual representation of sets Rectangle = Universal Set Sets are represented by circles

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Venn Diagrams U AB C A C B U

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Complement of a Set: If U is a universal set and A is a subset of U, then the set of all elements in U that are not in A is called the complement of A, written A C. Set Complementation Complement of a Set

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Set Operations Commutative Laws Associative Laws Distributive Laws

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. De Morgan’s Laws Let A and B be sets, then

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Given the sets: Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Venn Diagrams U A AB U

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Section 6.2 The Number of Elements in a Set The number of elements in a set A is denoted n(A). Ex. Given n(A) = 4 Notice Since the union doesn’t count a and h twice This leads to Overlap is subtracted

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Venn Diagram U A B – number of elements We can see that 22 So which leads to

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Survey In a survey of 100 people at a carnival: 40 like cotton candy 30 like popcorn 45 like lemonade 15 like lemonade and popcorn 10 like cotton candy and lemonade 12 like cotton candy and popcorn 5 like all three How many people don’t like lemonade, popcorn, or cotton candy? How many people only like popcorn?

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Section 6.3 The Multiplication Principle If there are m ways of performing a task T 1 and n ways of performing a task T 2, then there are mn ways of performing task T 1 followed by task T 2. Ex. A man has a choice of 8 shirts and 3 different pants. How many outfits can the man wear?

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. How many outcomes are possible for a game that consists of rolling a die followed by flipping a fair coin? 6 possibilities2 possibilities Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. An employee ID for a particular company consists of the employee’s first initial, last initial, and last four digits of his/her social security number. How many possible ID’s are there? Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Section 6.4 Permutations and Combinatons A permutation of a set of objects is an arrangement of these objects in a definite order. A combination is a selection of r objects from a set of n objects where order is not important

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. n–Factorial For any natural number n, Ex. 5! = 5(4)(3)(2)(1) = 120 This notation allows us to write expressions associated with permutations and combinations in a compact form. Ex.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Permutations of n Distinct Objects The number of permutations of n distinct objects taken r at a time is given by Ex.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be strung together in a row? Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Permutations of n Objects, Not all Distinct then the number of permutations of these n objects taken n at a time is given by Given n objects with n 1 (non-distinct) of type 1, n 2 (non-distinct) of type 2,…, n r (non-distinct) of type r where n = n 1 + n 2 + … + n r

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. How many distinguishable arrangements are there of the letters of the word initializing? Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Combinations of n Objects The number of combinations of n distinct objects taken r at a time is given by Ex. Find C(9, 6).

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex 1. A boy has 4 beads – red, white, blue, and yellow. How different ways can three of the beads be chosen to trade away? Example

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex 2. A space shuttle crew consists of a shuttle commander, a pilot, three engineers, a scientist, and a civilian. The shuttle commander and pilot are to be chosen from 8 candidates, the three engineers from 12 candidates, the scientist from 5 candidates, and the civilian from 2 candidates. How many such space shuttle crews can be formed? Solution:... Example