Enjoy - Finite Math - Quiz, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Download Enjoy - Finite Math - Quiz and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! MA 110-02 §1.1 – 2.4 Test #1 score Name: 26 June 2001 1. Use a properly labeled Venn diagram to determine the validity of the following argument. Explain. (10 points) 1. All politicians enjoy helping people. 2. Sue enjoys helping people. Therefore Sue is a politician. Solution: The argument is invalid. From the Venn dia- gram, we see Sue can enjoy helping while not being a politician. Politicians those who enjoy helping people Sue 2. Construct a truth table to show that the symbolic statement p → q is logically equivalent to its contrapositive. (10 points) Solution: p q p → q ˜q → ˜p T T T T T F F F F T T T F F T T The last two columns agree, so the statement p → q is logically equivalent to ˜q → ˜p. 3. Write the following argument in symbolic form. Then use a truth table to determine if the argument is valid. (10 points) If a student studies regularly, then the student does well in school. If the stu- dent’s teachers are good, then the student does well in school. The student does not do well in school. Therefore, the student doesn’t study regularly or the student’s teachers are not good. Solution: Use the symbolic representations p: a student studies regularly; q: the student does well in school; r : the student’s teachers are good. Then the paragraph can be written as ( (p → q)∧ (r → q)∧ (˜q) ) → ( ˜p ∨ ˜r ) which turns out to be a tautology, so the argument is valid. p q r p → q r → q ˜q (p → q)∧ (r → q)∧ (˜q) ˜p ∨ ˜r paragraph T T T T T F F F T T T F T T T T T T T F T F F F F F T T F F F T T F T T F T T T T F T T T F T F T T T T T T F F T T F F F T T F F F T T T T T T MA 110 Test 1 Solutions page 2 4. Which two of the following statements are logically equivalent? (You don’t need to use a truth table, but explain why they are in a sentence.) (10 points) (a) If it is not raining, then I play tennis. (b) If I play tennis, then it is not raining. (c) If it is raining, then I don’t play tennis. (d) I hate tennis, therefore I don’t play tennis. Solution: Statement (c) is the contrapositive of statement (b), so they are logically equivalent. Statement (b) is the converse of (a) so they aren’t equivalent. 5. If U = {a,b, c, d, e, f , g,h, i, j, k, l,m}, A = {a, c, d, g, j, k,m} and B = {a, c, e, g, i, k,m}, find the set (A ∩ B)′. Then illustrate (A ∪ B)′ by shading the result in a Venn diagram. (10 points) Solution: Beginning with the universal set U , just delete all elements that are in A ∩ B to get (A ∩ B)′ = {b,d, e, f , g,h, i, l}. The Venn diagram for (A∪B)′ is shown to the right. A B U 6. In a group of 250 students, 165 enjoy attending basketball games, 126 enjoy attending baseball games, and 61 enjoy neither? How many of the students enjoy both? Draw a properly labeled Venn diagram and explain your reasoning. (10 points) Solution: We know that n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Since we are given that 61 like neither, we know that 250− 61 = 189 like at least one, so n(A∪ B) = 189. So 189 = 164+128−n(A∩B). Thus, n(A∩B) = 102. 10263 61 Ba ske tball Baseball 24 7. Compute the numbers 7P3 and 7C3. Make up two counting problems that would have these numbers as an answer. (10 points) Solution: Using the formulas we have 7P3 = 7!(7−3)! = 7 · 6 · 5 = 210 and 7C3 = 7!(7−3)!3! = 7·6·53·2 = 35. For counting problems, you could say: From a group of 7 people, we want to select 3 and line them up in a row (answer 7P3). From a group of 7 people, we want to select 3 to form a committee (answer 7C3).