MTH 207 Final Sample, Fall 2012

1. (4 %) Show that (A � B) � C = A � (B � C) ; using any method 2. (6 %) Let f : Z � Z ! Z be de…ned by f(m; n) = 2m3n: (a) Is f 1-1? (b) Is f onto? 3. (5 %)Same question if f : R � R ! R � R 1 4. (5 %) Let � be a relation on Z given by m � n if and only if m3 = n3 Show that � is an equivalence relation and …nd the equivalence classes 5. (6 %) Fill in the blanks with the appropriate symbols (2; =2;or �; ") in the following cases: (a) 1:::::f1; 2; f1g; f1; 2gg (b) f2g::::f1; 2; f1g; f1; 2gg (c) ff1gg:::f1; 2; f1g; f1; 2gg (d) f1; f2gg:::f1; 2; f1g; f1; 2gg 6. (9%) If you know that the proposition (p^ � (q^ � s)) ! (p ^ r) is false, what can you say about the truth value of the propositions (a) q _ r (b) q ! s (c) r _ (q _ s) 7. (3 %) Write the negation of the following statements (a) All students are given a second chance (b) If we relax, we will perform (c) If the weather permits and if we are lucky then we will meet in the park 2 8. (10%) Consider the following digraph: (a) Write the matrix that represents the graph (b) Find a path of length 3 that goes from y to w: (c) Find a path of length 3 that goes from v to w: (d) Find a path of length 4 that goes from v to y: (e) Find a path of length 3 that goes from z to itself: 9. (6%) Write your own relation on the set f1; 2; 3; 4; 5g that is (Re‡exive) and (Anti Symmetric) and (Transitive).Draw its digraph. 10. (6%) Write an explicit formula;a for an = 3an�1 � 2an�2; n � 3; where a1 = 5; and a2 = 3: 3 11. (7%) If bn = 4 + bn�1; for all n � 1; if b0 = 5;show that bn = 5 + 4n 12. (7%) Consider the matrices: A = 2 4 4 5 �3 2 1 4 3 1 0 3 5 and B = 2 4 2 �1 1 0 2 2 1 0 3 3 5 : Evaluate the following: (a) A + B (b) 2A � 3B (c) A:B (d) B:A (e) At 4 13. (7%) Prove by induction that n(n2 + 5) is divisible by 6. 14. (6%) A license plate for a car consists of 3 letters followed by 4 digits. (a) How many possible licence plates are there? (b) How many are there with no repeated letters and non repeating digits? (c) What is the probability that the license plate has 3 repeated letters? 15. (6%) A die is rolled twice (a) What is the probability of getting a least one even number on one of the dice? (b) What is the probability of getting a double? 5 (c) What is the probability of getting a sum of 9. 16. (7%) A study was done on a certain drugs, A, B, and C. It was shown that out of 40 people, 23 reported relief from drug A, 18 reported relief from drug B, 31 reported relief from drug C, 11 reported relief from drug A and B, 19 reported relief from drug A and C, 14 reported relief from drug B and C, and 37 reported relief from at least one of the drugs. (a) How many reported relief from none of the drugs? (b) How many reported relief from all three drugs? 6