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7.1 Sets This presentation is copyright by Paul Hendrick © 2003-2007, Paul Hendrick All rights reserved

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7.1 Sets Definitions & Examples A set is a well-defined collection of objects –The objects in a set are called its members or elements –Sets are usually named by capital letters, such as A, B, X, Y, etc. –(If the elements of a set are letters, it’s usually best to use lower-case for these.) The English alphabet: A = {a, b, c, …, z} –(note: 26, not 52; i.e., capital & l.c. not considered different) Be sure to use the set brackets { }

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7.1 Sets Definitions & Examples (cont.) –Example: let F={1,2,3,4,5}, the first five counting #s. –Example: let P = {x | x is a prime # and x 10} P could also be given as follows: P={2,3,5,7} Set-builder notation vs. roster notation –Sets A and B are equal iff they contain the same elements F={1,3,5,4,2} same as original F Order doesn’t matter. F={1,2,3,4,5,4,3,2,1} same as original F! –Note: still only 5 elements! Repeats don’t count. –The last is considered bad form! (It’s misleading.) –DON’T do this!

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The Universal set, , in any given situation, is the set of all possible elements from which elements for a set could be taken. –E.g., if we want set B to be the set of students taking this course and section, then could be the set of all Blinn math students at the Bryan campus, or all Blinn students at any campus, or all students in the world. –E.g., if we want the set C = some set of cards in a poker hand, we would have to change (it’s a different situation!) 7.1 Sets Definitions & Examples (cont.)

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The Empty set is the set that has no elements. Also called the null set –Designated either by { } or by Don’t write it as { } !! (that’s a set with 1 element!) –Let W = the set of all students who cut all the major exams and make an “A” in this class. W = . –Let X = the set of all aliens (extra-terrestrial) living amongst us. X = { }. –This is the instructor’s personal opinion! 7.1 Sets Definitions & Examples (cont.)

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A set A is a subset of a set B iff every element of A is also an element of B. We write A B (note the _, as an “equal” possibility, similar to x 4 or 4 4 ) –A set A is a proper subset of a set B iff A is a subset of B, but A is not the same as B. We write A B (note NO _, as an “equal” possibility, similar to x < 4 or 3 < 4 ) –E.g., recall F={1,2,3,4,5} and P={2,3,5,7} –If T={1,2,3,…,9,10}, then F T or F T, but F P –Note: A B and B A iff A = B.

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7.1 Sets Definitions & Examples (cont.) –Don’t confuse elements and subsets! –Recall P={2,3,5,7} 2 P {2} P 2 P {2} P P –The null set is a subset of every set! P Y –Every set is a subset of the Universal set!

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7.1 Sets Definitions & Examples (cont.) 7.1 #54 (p. 322) Let B={a,b,c,{d},{e,f}}. True or false each? {a} B –False a is an element of B, but {a} is not {b,c,d} B –False b and c are elements of B, but d is not; {d} is. {d} B –True {d} B –False {d} is an element of B, but d is not {e,f} B –True {a,{e,f}} B –True {e,f} B –False {e,f} is an element of B, so {{e,f}} IS a proper subset

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7.1 Sets Definitions & Examples / Power Set How many subsets does the set {a,b,c,d} have? Subsets of size 0: { } 1 Subsets of size 1: {a}, {b}, {c}, {d} 4 Subsets of size 2: {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d} 6 Subsets of size 3: {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d} 4 Subsets of size 4: {a,b,c,d} 1 –16 subsets in all (15 proper subsets) –The set { { }, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d} } is called the power set for the set {a,b,c,d}. –The set of all possible subsets of a set A is called the POWER SET for the set A, and is designated by 2 A. –If n(A) designates the size of set A, then n(2 A ) = 2 n(A)

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7.1 Sets Tree Diagrams –Tree diagrams can be useful for illustrating various processes for probability problems –Branches connect nodes in the tree diagram A B E D C G F H I J K A is a root node B is a branch node D is a terminal node AB is a branch GK is a branch ABEI is a path (could also be called AI) 6 – the same as # of terminal nodes! How many paths in this tree?

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7.1 Sets Tree Diagrams –Tree diagrams can be used to illustrate the building of a subset –(illustration of tree diagram on board) –Each branch in the tree diagram represents a choice to include or not include a given element in the subset –Each path in the tree diagram represents all the choices made, over all of the original elements –If a path has n choices to be made, with 2 outcomes on each choice, then there are 2 n possible ways to make all the choices.

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7.1 Sets Set operations –Complement –Union –Intersection

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7.1 Sets Set operations -- Complement –The complement of a set A is the set of all elements that are not in A (but which, of course, are in ). In symbols, A = {x | x and x A} Since it’s understood that every element is in , usually don’t bother to say it! A = {x | x A}

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7.1 Sets Set operations -- Union –The union of two sets A and B is the set of all elements that are in A or are in B (or are in both). This is called the “inclusive or”. (Of course, all are in U, so we don’t need to bother to say anything about that!) A B = {x | x A or x B } –E.g., recall F={1,2,3,4,5} and P={2,3,5,7} F P = {x | x F or x P } = {1,2,3,4,5,2,3,5,7} = {1,2,3,4,5,7}

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7.1 Sets Set operations -- Intersection –The intersection of two sets A and B is the set of all elements that are in A and are in B (at the same time). A B = {x | x A and x B } –E.g., recall F={1,2,3,4,5} and P={2,3,5,7} F P = {x | x F and x P } = {1,2,3,4,5} {2,3,5,7} = {1,2,3,4,5} in F, less what’s not in P = {2,3,5} //

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7.1 Sets Set operations – Union and Intersection –Recall that for F={1,2,3,4,5} and P={2,3,5,7} F P = {1,2,3,4,5,7} F P = {2,3,5} –Note that F P F P –For any two sets A and B, A B A B Can the intersection and union ever be equal? When?

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7.1 Sets Set operations -- Intersection –If the intersection of two sets A and B is empty, that is, the two sets have nothing in common, then the sets are said to be disjoint. E.g., let E = {2,4,6,8,10} (even integers in T), and let O = {1,3,5,7,9} (odd integers in T), then E and O have no elements in common. E O = E and O are disjoint.

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7.1 Sets Set operations vs words –Complement vs. “not” A = {x | x , x is not an element of A} –Union vs “or” A B = {x | x A or x B } –Intersection vs. “and” A B = {x | x A and x B }

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7.1 Sets Definitions & Examples (cont.) Venn diagrams are useful in representing sets, subsets, and operations with sets –These were invented by the English logician John Venn –These consist of circles representing sets inside of one rectangle representing the Universal set –The inside of a circle represents the contents of the set –A subset of another set is shown by a circle within another circle –(illustration of Venn diagrams on board)

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Chapter 2 The Basic Concepts of Set Theory

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