lecture03 - MA1100 Lecture 3 Sets Set Operations Indexed...

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1 MA1100 Lecture 3 Sets Set Operations Indexed Collection of Sets Partitions of Sets Cartesian Products of Sets Chartrand: section 1.3 – 1.6
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Lecture 3 2 Complement Let A be a subset of a universal set U. The complement of A is the set of all elements of U that are not in A . Notation : A c Set notation: { x œ U | x A} Example A = { x œ R | x < 3 } A c = {x œ R | x NOT less than 3 } If U = R , then Q c = or A (in Chartrand book) = { x œ R | x Q }
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Lecture 3 3 Complement U c = { x œ U | x U } Let U be some universal set. No such element exists. « c = { x œ U | x –« } Every element of U are not in the empty set .
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Lecture 3 4 Relative Complement Let A and B be subsets of a universal set U. The relative complement of B with respect to A is the set of all elements that are in A but not in B . Notation : A – B Set notation: { x œ U | x œ A and x B } Example A = { 1, 2, 3, 4 } B = { 3, 4, 5, 6 } A – B = (or A \ B ) For any set A with universal set U, A c = U – A ( Chartrand also calls this set difference )
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Lecture 3 5 Relative Complement = { x œ R | x œ A and x B } Example A – B A = { x œ R | x < 3 } B = { x œ R | x > -2 } = { x œ R | x < 3 and x not greater than -2 }
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Lecture 3 6 Venn Diagrams of Set Operations U Venn diagram B A 1 2 3 4 Set region A B A » B A c A – B
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Lecture 3 7 Venn Diagrams of Set Operations (i) Indicate in the two Venn diagrams the regions representing (A » B) (C » D) and (A C) » (B D). (ii) Is (A » B) (C » D) = (A C) » (B D) ? AB C D C D
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Lecture 3 8 Limitation of Venn Diagrams A B CD
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Lecture 3 9 Indexed Collection of Sets Example { 1 } Sometime we may need to consider collection of sets that are “related” or “similar”. A 1 A 2 A 3 In general, A n = { 1, 2, 3, …, n} { 1, 2 } { 1, 2, 3 } { 1, 2, 3, 4 } … [ 1, )[ 2 , 3 , 4 , ) … A 4 B 1 B 2 B 3 In general, B n = [ n, ) B 4 = { x œ N | x § n } = { x œ R | n § x < }
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Lecture 3 10 Intersection (Indexed Collection ) Let A 1 , A 2 , A 3 , … be an indexed collection of sets.
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture03 - MA1100 Lecture 3 Sets Set Operations Indexed...

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