Announcements ICS 6B Dont forget Regrades for Quizzes 1 & 2, - PDF document

Announcements ICS 6B � Don’t forget Regrades for Quizzes 1 & 2, and Homeworks 1‐3 are due today Boolean Algebra & Logic Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 6 – Ch. 8.4 2 Lecture Set 6 - Chpts 8.3 Where you stand Quizzes � Homeworks � Quiz #1 12 – 90‐100 ● Max: 97% ● Min: 28% 4 ‐ 80‐89 ● Avg: 70% 2 ‐ less than 5 � Quiz #2 ● Max: 97% ● Min: 46% ● Avg: 77% Lecture Set 6 - Chpts 8.3 3 Lecture Set 6 - Chpts 8.3 4 Some perspective Overall You can drop your lowest Quiz Score 3 – 90‐100 3 – 80‐89 I suffer from test anxiety – what can I do? 3 – 70‐79 http://www.studygs.net/tstprp8.htm 4 – 60‐69 http :// ub‐counseling.buffalo.edu/stresstestanxiety.shtml // 1 – 50‐59 http://www.sdc.uwo.ca/learning/mcanx.html 4 – less than 50 http://www.kidshealth.org/teen/school_jobs/school/ test_anxiety.html Lecture Set 6 - Chpts 8.3 5 Lecture Set 6 - Chpts 8.3 6 1

How do I improve my performance on Today’s Lecture Quizzes – and the final? � Chapter 8 �8.3, 8.4� � If you have to miss lecture – ● Representing Relations �8.3� ● get notes from your friends ● Closures of Relations �8.4� � Review lecture slides �take notes� � Do the reading � Form a study group y g p � Ask questions ● In class ● Email ● Office hours � What if I aced them? � WTG! 7 8 Lecture Set 6 - Chpts 8.3 Lecture Set 6 - Chpts 8.3 Property Definition Matrix Def In other words Properties on Relations Reflexive �i�1…n �i�1…n All the diagonal entries of M R =1 (a i , a i ) � R m ii =1 Irreflexive �i�1…n �i�1…n All the diagonal entries of M R =0 � (a i , a i ) � R Chapter 8: Section 8.3 m ii =0 �i,j�1…n, i�j �i,j�1…n, i�j Symmetric m ij ,= m ji is either (0,0) or (1,1) (a i , a j ) � R & � a j , m ij ,= 1 & m ji ,= 1 M R is symmetric wrt diagonal a i ) � R m ij ,= m ji Antisymmetric �i,j�1…n, i�j �i,j�1…n, i�j {m ij ,,m ji } ≠ {1,1} (a i , a j ) � R & � a j , R & � ( ) m ij ,= 1 & m ji ,= 0 1 & 0 a i ) � R m ij ≠ m ji If i=j then {1,1} or {0,0} Representing Relations Asymmetric R both irreflexive �I,j�1…n All the diagonal entries of M R =0 & m ii =0 AND i�j Antisymmetric m ij ,,m ji } ≠ {1,1} m ij ≠ m ji If i=j then {1,1} or {0,0} Transitivity �i,j,k�1…n Well discuss this If (a i , a j ) � R & later (a j , a k ) � R then (a i , a k ) � R Lecture Set 6 - Chpts 8.3 10 Transitivity revisited Transitive Property Example Consider the relations on �1,2,3,4� Consider the relations on �1,2,3,4� Which are Transitive? � R 5 ���1,1�,�1,2�,�1,3�,�2,1�,�2,2�,�2,3�,�3,3�,�4,4�� R 1 ���1,1�,�1,2�,�2,1�,�2,2�,�3,4�,�4,1�,�4,4�� R 2 ���1,1�,�1,2�,�2,1�� x�y y y�z y x�z 1 2 R 3 ���1,1�,�1,2�,�1,4�,�2,1�,�2,2�,�3,3�,�4,1�,�4,4�� R ��1 1� �1 2� �1 4� �2 1� �2 2� �3 3� �4 1� �4 4�� �2, 1� �1, 3� �2, 3� R 4 ���2,1�,�3,1�,�3,2�,�4,1�,�4,2�,�4,3�� �1, 2� �2, 3� �1, 3� R 5 ���1,1�,�1,2�,�1,3�,�2,1�,�2,2�,�2,3�,�3,3�,�4,4�� 3 4 4, 5,6 R 6 ���3,4�� 4- shown in slide set #5 5 - shown in previous slide 6 - because (a, b) � (b,c) is false so (a, b) � (b,c) � �c,d� is true Lecture Set 6 - Chpts 8.3 11 Lecture Set 6 - Chpts 8.3 12 2

Using Matrices to represent Relations Matrices Examples R2���1,1�,�1,2�,�2,1�� Express the relation R 1 as a matrix Express the relation R 1 as a matrix R3���1,1�,�1,2�,�1,4�,�2,1�,�2,2�,�3,3�,�4,1�,�4,4�� A��1.2.3.4�. B��1,2,3,4� A��1.2.3.4�. B��1,2,3,4� R4���2,1�,�3,1�,�3,2�,�4,1�,�4,2�,�4,3�� R1���1,1�,�1,2�,�2,1�,�2,2�,�3,4�,�4,1�,�4,4�� R1���1,1�,�1,2�,�2,1�,�2,2�,�3,4�,�4,1�,�4,4�� R5���1,1�,�1,2�,�1,3�,�2,1�,�2,2�,�2,3�,�3,3�,�4,4�� Think of the �a, b� pairs and remember that A�Rows Think of the �a, b� pairs and remember that A�Rows and B� Cols and B� Cols B B 0 [ ] M M R3 M M R4 M M R5 [ ] No [ ] No [ ] M R2 M [ ] 1 2 3 4 1 1 0 1 0 0 0 0 1 1 1 0 1 �1,1� �1, 2� �1,3� �1,4� �1,1� �1, 2� �1,3� �1,4� 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 2 �2,1� �2, 2� �2,3� �2,4� �2,1� �2, 2� �2,3� �2,4� A 1 1 0 0 M R1 � 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 3 �3,1� �3, 2� �3,3� �3,4� �3,1� �3, 2� �3,3� �3,4� 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 4 �4,1� �4, 2� �4,3� �4,4� �4,1� �4, 2� �4,3� �4,4� Reflexive? No Yes No Yes Is R1 Reflexive? No Symmetric? Yes Yes No No Put a 1 in each of the Is R1 Symmetric? No Antisymmetric? No No Yes No Corresponding locations No Irreflexive? Yes No Is R1 Antisymmetric? 13 14 Lecture Set 6 - Chpts 8.3 Lecture Set 6 - Chpts 8.3 Operations on Matrices/Relations Join & Meet – bit operations Let R & S be relations on a set A with or � These are a “join” b � a � b 1 if a�1 � b�1 corresponding matrices M r & M s defined in 0 otherwise more detail and � in section How do we find ? 1 if a�1 � b�1 a “meet” b � a � b 3.8 0 otherwise M r�s � M r � M s & M � M � M & In other words.. M r �s � M r � M s a � b � 0 if a�b�0, otherwise it is 1 a � b � 1 if a�b�1, otherwise it is 0 It helps if we first introduce a few operations on bits then apply it to Now we can extend this to matrices matrices…. (term by term) Lecture Set 6 - Chpts 8.3 15 Lecture Set 6 - Chpts 8.3 16 Join and Meet - Matrices Boolean Product For any two binary �n x n� matrices A & B Let A and B be nxn, 0-1 matrices. The boolean product is a nxn, 0-1 matrix We define A � B and A�B to be the n x n binary whose ij element is matrices whose ij element is given by: (a i1 � b 1j ) � (a i2 � b 2j ) � … � (a in � b nj ) �A join B� ij � A ij � B ij & �A meet B� ij � A ij � B ij , respectively �A meet B� � A � B respectively Notation: A� B A B A � B A � B 1 0 [ ] In other words: [ ] 0 0 [ ] [ ] 1 1 1 0 1 � 1 1 � 0 1 1 � � � 0 � 1 0 � 0 1 0 ● �A� B� ij �1 iff at least 1 of the terms A B A � B A � B a ik � b kj �1 [ ] 0 0 [ ] 1 0 [ ] [ ] 1 1 1 0 1 � 1 1 � 0 1 0 or �k�a ik �b kj �1� � � � 0 � 1 0 � 0 0 0 Lecture Set 6 - Chpts 8.3 17 Lecture Set 6 - Chpts 8.3 18 3