Sets Large and Small

7/9/2013 Sets 2 2 Set Definition: A set is a collection of objects, real or virtual, with a clear defining property for membership in the set Examples: The set of all points on circle x 2 + y 2 = 1 The set of all tires on your car The set of all left shoes owned by you The set of all integers in interval (0, 10) The set of all pink elephants under your bed

7/9/2013 Sets 3 List notation: { 4, 2, D, 5, A, H, 7, -3 } Set-builder notation: { x | x is an integer } Sets are not ordered Repeated elements ignored Sets 33 Notation Note: a { a } for any object a : { 4, 2, D } = { D, 2, 4 } : { 4, 4 } = { 4 } 1 { 1 }, 0 { 0 }, etc.

7/9/2013 Sets 4 Set Membership If x is an element of set A we write If x is NOT an element of set A we write Sets 44 Notation Note: c A x A x c To avoid paradoxes we disallow sets A such that c A A * * See the notes page

7/9/2013 Sets 5 5 Building New Sets Set Operations Union The union of sets A and B is the set of all elements in A or B or both, written Intersection The intersection of sets A and B is the set of elements in both A and B A B A B

7/9/2013 Sets 6 6 A B = { 0 } Set Operations Examples 1. A = { m, n, k, 3, 5, x, y, z } B = { j, k, 2, 3, 4 } 2. A = { x | x 0 } B = { x | x 0 } A B = { m, n, j, k, 2, 3, 4, 5, x, y, z } A B = { k, 3 } A B = R

7/9/2013 Sets 7 7 Set A Set B Set Operations Venn Diagrams A B Set A Set B A B A B = { x | x A or x B } A B = { x | x A and x B }

7/9/2013 Sets 8 Set Within a Set If for each This can also be written If A B then A is a proper subset of B If A = B then Sets 8 Subsets A B x A c x B c it is true that then A is a subset of B, written BA and A B B A

7/9/2013 Sets 9 Whats NOT in the Set The complement of Set A is relative to universe of discourse U The complement of set A relative to B Sets 99 Set Complement = { x | x A } c AcAc = ~A A – B = A \ B= A / B { x | x A, x B } = c c

7/9/2013 Sets 10 Sets 10 The Natural Numbers The Set of Counting Numbers N = { 1, 2, 3, 4, … } The Integers Positive and Negative Natural Numbers I = { … -3, -2, -1, 0, 1, 2, 3, … } Subsets of the Real Numbers R R

7/9/2013 Sets 11 Sets 11 Rational Numbers Solutions of linear equations a x + b = 0 for integers a, b Proper fractions and integers Subsets of the Real Numbers Q = { | a, b are integers, b 0 } a b

7/9/2013 Sets 12 Sets 12 Irrational Numbers Not solutions of a x + b = 0 Algebraic numbers – roots of n th degree polynomials with rational coefficients Examples: x 2 – 2 = 0 x 3 – 5 = 0 Subsets of the Real Numbers { x | x R, x Q } x = 2 + – x = 3 5 R

7/9/2013 Sets 13 Sets 13 Irrational Numbers Not solutions of a x + b = 0 Transcendental numbers – Examples: = … e = … Subsets of the Real Numbers { x | x R, x Q } NOT roots of any polynomial with rational coefficients R

7/9/2013 Sets 14 The Set With No Members The empty set, or null set, is written There is Thus Sets 14 The Empty Set { } O OR no x such that x c O = O { x | x > 0 and x < 0 } = { x | x = pink elephant under your bed } = { x | x is rational and x is irrational } Note: O { } O for all sets A O A

7/9/2013 Sets 15 Sets 15 Ordered Pairs Sets of Ordered Pairs Notation: Example: Colors Ordered Pair: { ( a, b ), ( c, d ), ( e, f ), ( g, h ) } Set of ordered pairs: { ( orange, blue ), ( red, green ), … } ( red, green )

7/9/2013 Sets 16 Sets 16 Ordered Pairs Example: Numbers Ordered pair: Set of ordered pairs: { ( -6, 10 ), ( 3, 0 ), ( 2, 7 ), ( -2, 0 ) } ( 3, 0 ) Question: Does the order of the pairs matter ?

7/9/2013 Sets 17 Sets 17 Ordered Pairs Numbers and Colors Ordered pair: Set of ordered pairs: { ( 0, black ), ( 1, red ), ( 2, yellow ) … } ( 3, blue ) Example: Question: Does this appear to be a function ?

7/9/2013 Sets 18 Sets 18 Think about it !

7/9/2013 Sets 19 Sets 19 Spare Parts c c c c c cc A – B = A \ B = A / B = { x | x A, x B } c A B c x A c x A c x B c AB A B B A c c cc ccc cc