No books, notes, calculators, or collaboration. 1) (14 pts) Provide a justification (rule and line numbers) for each line of this proof:

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1 Family name Given name ID# Math 2000A Final Exam (Fall 2012) Time limit: 2 1 hours 2 There are 200 total points on this test. No books, notes, calculators, or collaboration. 1) (14 pts) Provide a justification (rule and line numbers) for each line of this proof: 1 (u M) (u / P Q) 2 (u P ) (u R) 3 (u / P Q) (u M) 4 (u P Q) (u / M) 5 u M N 6 u P 7 (u P ) (u Q) 8 u P Q 9 u / M 10 (u M) & (u N) 11 u M 12 (u M) & (u / M) 13 u / P 14 u R 15 (u M N) (u R) 2) (10 pts) Assume D is a set. Circle T if the assertion is always true (a tautology). Circle F if the assertion is always false (a contradiction). Circle? if the assertion is sometimes true and sometimes false (depending on what D is). (You do not need to show your work.) SCORE = # of correct answers minus # of wrong answers. T F? x, (x = x) T F? x, (x x) T F? x, (x = x) T F? x, (x x) (Parts left blank will be ignored.) T F? d D, (d D) T F? d D, (d D) T F? d D, ( (d / ) (d ) ) T F? d D, ( (d ) (d / ) ) T F? d D, ( (d / ) (d ) ) T F? d D, ( (d ) (d / ) )

2 3) (8 pts) Define a sequence {g n } by g 1 = 5, and g n = 2g n 1 3n + 6 for n 2. Prove by induction that, for every n N +, we have g n = 2 n + 3n. 4) (10 pts) Give a two-column proof of the following deduction: ( p Q R ) ( p M (T S) ),.. (p Q) (p / S).

3 5) (10 pts) Explain how you know that the following deduction is not valid: D E = F, G E = H,.. D G F H. 6) (5 pts) Assume (a) is an equivalence relation on R, and (b) for every x, y R, if xy = y, then x y. Show 3 4.

4 7) (12 pts) Indicate whether each of the following sets is countable (C) or uncountable (U). (Circle the correct response for each set.) You do not need to justify your answers! Score = # correct answers # wrong answers. (Items left blank will be ignored.) C U R N C U Q N C U (R Q) (Q R) C U (R Q) (Q R) C U (R Q) (Q R) C U (R Q) (Q R) C U { x R x 2 6 < 0 } C U { (x, y) R Z x + y Q } C U { x R x < 0 } C U { (x, y) R Z x + y R } C U { x R x 2 6 Q } C U { (x, y) Q Z x + y R } 8) (8 pts) Prove by induction that, for every n N +, we have n (10k 3) = 5n 2 + 2n. k=1

5 9) (10 pts) Assume E, F, and G are sets. Show (E F ) (G G) (E G) F. 10) (10 pts) Show that if P Q R S and R P Q, then Q S =.

6 11) (15 pts) Define j : R R by j(t) = 7t 12. Prove (directly from the definitions) that j is a bijection. Notation. Suppose f : A B, A 1 A, and B 1 B. Then f(a 1 ) = { f(a) a A 1 } and f 1 (B 1 ) = { a A f(a) B 1 }. 12) (10 pts) Assume h: X Y and X 1 X. Show (directly from the definitions) that if h is one-to-one, then h 1( h(x 1 ) ) X 1.

7 13) (10 pts) Assume p, q, r Z. Show (directly from the definitions) that if p q (mod 4r) and p 2q (mod 6r), then p 0 (mod 2r). 14) (10 pts) Assume h: D D, and h h = h. Show that if h is onto, then h is a bijection.

8 15) (4 pts) Simplify (15 30) ( ) in Z 9. Show your work! 16) (4 pts) Simplify( the assertion ((r ) ( ) ) r R, B) (r a) & b B, (b = a) (b = r) Show your work! 17) (4 pts) Assume (a) E is a binary relation on R, and (b) for every x, y R, if xy = 100, then x E y. (That is, we have (x E y).) Show E is not an equivalence relation.

9 18) (4 pts) Assume P is true, but Q and R are false. Calculate the truth value of the assertion ( P (Q R) ) & ( (P Q) R ). Show your work! 19) (10 pts) Short answer. (You do not need to answer in complete sentences, and you do not need to show your work.) (a) (2 pts) Specify the following power set by listing its elements: P ( {,, } ) = (b) (2 pts) What does it mean to say that a set S is countable? (c) (2 pts) What is the usual way to prove that two sets are equal? (d) (2 pts) What does it mean to say that a set S is finite? (e) (2 pts) Assume S = {0, 1, 2, 3, 4}. { (s1, s 2 ) S S s1 + s 2 = 3 } = Specify the following set by listing its elements:

10 20) (18 pts) Short answer. (You do not need to answer in complete sentences, and you do not need to show your work.) (a) (2 pts) What is the Law of Excluded Middle? (b) (2 pts) What does it mean to say that a set T is countably infinite? (c) (3 pts) What is the Pigeonhole Principle? (d) (4 pts) What does it mean to say that c is a function from H to Q? (e) (3 pts) State the converse and the contrapositive of the following assertion: If the sun is shining, then I am happy. Converse: Contrapositive: (f) (2 pts) Let = {(1, 1), (1, 4), (2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (4, 1), (4, 4), (5, 2), (5, 3), (5, 5)}, so is an equivalence relation on {1, 2, 3, 4, 5}. (You do not need to prove this fact.) Specify the following set by listing its elements: [3] = (g) (2 pts) If S is a set and t N, what does it mean to say that #S = t?

11 21) (14 pts) Short answer. (You do not need to answer in complete sentences, and you do not need to show your work.) (a) (2 pts) What does it mean to say that the set S has the same cardinality as the set T? (b) (3 pts) Suppose a: S T. What does it mean to say that b: T S is the inverse of a? (c) (2 pts) Draw a Venn diagram of the set ( A (B C) ) (C A). (d) (4 pts) Using the symbolization key U: the set of all people T : the set of all tall people S: the set of all smart people H: the set of all happy people R: the set of all rich people translate the following assertion into the notation of First-Order Logic: If every happy person is either tall or not rich, then no smart person is both happy and rich. (e) (3 pts) What is the Well-Ordering Principle?

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