lecture03&iuml;&frac14;ˆcomplete&iuml;&frac14;‰

# lecture03&iuml;&frac14;ˆcomplete&iuml;&frac...

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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 Announcement ± Tutorial balloting close tomorrow (NUS students only) ± Lecture quiz buddy (LQB) sign up form (for those who yet to find a buddy) ± Collection of lecture quiz clickers ± Mock-up lecture quiz for this week. ± Introductory handouts
Lecture 3 3 Œ vs § (an analogy) 2 § 5 2 < 5 2 = 5 5 § 5 5 < 5 5 = 5 {a, b} Œ {a, b, c} {a, b} Õ {a, b, c} {a, b} = {a, b, c} {a, b, c} Œ {a, b, c} {a, b, c} Õ {a, b, c} {a, b, c} = {a, b, c}

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Lecture 3 4 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: A c = { 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 = = I , the set of irrational numbers or A (in Chartrand book) = { x œ R | x ¥ 3 } = { x œ R | x Q } U A A c
Lecture 3 5 Complement U c = { x œ U | x U } Let U be some universal set. No such element exists. So U c = « « c = { x œ U | x –« } Every element of U are not in the empty set . So « c = U So U c must be empty . Every element of U satisfies x . What are U c and « c ? A c = { x œ U | x A }

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Lecture 3 6 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: A - B = { x œ U | x œ A and x B } Example A = { 1, 2, 3, 4 } B = { 3, 4, 5, 6 } A – B = { 1, 2 } (or A \ B ) For any set A with universal set U, A c = U – A ( Chartrand also calls this set difference ) U B A A – B
Lecture 3 7 Relative Complement = { x œ R | x œ A and x B } = { x œ R | x < 3 and x not greater than -2 } Example A – B A = { x œ R | x < 3 } B = { x œ R | x > -2 } = { x œ R | x < 3 and x § -2 } = { x œ R | x § -2 } redundant

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Lecture 3 8 Venn Diagrams of Set Operations U Venn diagram B A 1 2 3 4 Set region A B A » B A c A – B 2 1+2+3 3+4 1 3 B – A A c B c 4
Lecture 3 9 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) ? Ans: No Example: Take A = D = {1}, B = C = {2}. AB C D C D (A » B) (C » D) = {1, 2} (A C) » (B D) = « LQ1 Anything wrong?

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Lecture 3 10 Limitation of Venn Diagrams Suppose x œ A, x œ D, x B, x C.
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lecture03&iuml;&frac14;ˆcomplete&iuml;&frac...

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