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Unformatted text preview: MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 4 Solutions Spring 2010 3.1 : 2, 3, 8, 12, 15, 16; 3.2 : 2, 3, 7, 12; 3.3 : 2, 6, 18, 19, 20, 21. (Some proofs are given in several versions.) 3.1.2 Consider the Theorem: Suppose that b 2- 4 ac > 0. Then the quadratic equation ax 2 + bx + c = 0 has exactly two real solutions. (a) Hypothesis: a , b , and c are real numbers satisfying b 2- 4 ac > 0. Conclusion: there exists exactly two real solutions x to the equation ax 2 + bx + c = 0. (b) To give an instance of the theorem, we must specify a , b , and c , because we must show that they satisfy the hypothesis. We dont have to give a value for x because the existence of x is the conclusion, i.e., is asserted by the theorem. (c) When a = 2, b =- 5, and c = 3, we notice that b 2- 4 ac = 25- 4(2)(3) = 1 > 0, so the theorem asserts the existence of exactly two solutions to the equation 2 x 2- 5 x +3 = 0. Since 2 x 2- 5 x +3 = (2 x- 3)( x- 1), we see by direct calculation that the two roots are x = 3 / 2 and x = 1. (d) In the case a = 2, b = 4, and c = 3, we see that b 2- 4 ac = 16- 24 =- 8 < 0, so the hypothesis is not satisfied. Therefore we can draw no conclusion whatsoever. 3.1.3 Consider the Incorrect Theorem: Suppose n is a natural number larger than 2, and n is not prime. Then 2 n + 13 is not prime. Identify the hypotheses and conclusion, and show the theorem is incorrect by finding a counterexample. Solution: Hypotheses: n is a natural number, and n 2, and n is not prime. Conclusion: 2 n + 13 is not prime. Counterexample: Let n = 8, which is a natural number, 8 2, and 8 is not prime. But 2(8) + 13 = 29 is prime, therefore the Theorem is false. 3.1.8 (3 points) Suppose A \ B C D and x A . Prove that if x / D , then x B . Proof 1 (Direct proof): Assume that A \ B C D , that x A , and that x / D . Then x / C D , since C D D . But then x / A \ B , since A \ B C D . This says that x / A or x B . Since we are given that x A , it must be true that x B . This completes our proof. Proof 2 (Indirect proof): Suppose that A \ B C D and x A . Assume that x / B . We wish to show that x D . Now, we are assuming that x A and that x / B . But this means that x A \ B . So, by assumption, we conclude that x C D . This means that x C and x D . We have proved that x D . 3.1.12 (3 points) Suppose x and y are real numbers and 3 x + 2 y 5. Prove that if x > 1, then y < 1. Proof 1 (Direct proof): We are given that x and y are real numbers satisfying that 3 x + 2 y 5. Suppose that x > 1. Then, from 1 < x , we conclude that 3 < 3 x . But we are given that 3 x + 2 y 5, so 3 x 5- 2 y . This means that 3 < 5- 2 y , or 2 y < 5- 3 = 2. From this we conclude that y < 1, as desired....
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- Spring '07