<p>Discrete Structures (COSC 208) Exam 1 Time: 50 minutes Total points: 100 Student Name:</p><p>1. State which of the following are propositions and which are not? If it is a proposition then state its truth value. If not then explain why not. (10 pts) a) If 1  0, then 3  4.</p><p> b) x(x 2  0)</p><p> c) xy(x  y  z)</p><p> d) Is DC the capital of United States?</p><p>2. Let p and q be the propositions p : you are a juniors q : you get a scholarship write the following propositions using p and q and logical connectives (operators): (10 pts)</p><p> a) Being a junior is sufficient for getting a scholarship.</p><p> b) You will get a scholarship only if you are a junior. 3. Write out the truth tables for the following Boolean operators: (20 pts) a)  b)  c)  (this is xor) d)  e) </p><p>4. Find a proposition with the given truth table. (10 pts)</p><p> p q ? T T F T F F F T T F F F 5. Determine whether (p  q)  (p  q)  q. (You can use truth tables) (10 pts)</p><p>6. Show that ( p  q)  ( p  q) is a tautology without using truth tables. (10 pts) 7. In the questions below P(xy) means “x and y are real numbers such that x  2y  5”. Determine whether the statement is true or false. Explain why it is true or false. Give a solution or a counter-example where necessary. (10 pts)</p><p> a) xyP(xy).</p><p> b) xyP(xy).</p><p>8. In the questions below suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(xy): student x is taking course y. Write the statement using these predicates and any needed quantifiers. (20 pts)</p><p> a) Eric is taking MTH 281.</p><p> b) All students are freshmen.</p><p> c) Every freshman is a full-time student.</p><p> d) No math course is upper-level. (Bonus Problem) 9. Express the statement “All students in the class are less than 50 years old” using predicates and quantifiers with a domain consisting of all people. (10 pts)</p>