MATH 211 Exam 1, Fall 2005

MATH 211 EXAM ONE Fall 2005 September 29 Name: The test consists of 2 pages. Justify your work when necessary. (1) Find a simplifying expression for the following sets. Here U is the universe, A and B are two sets such that B ⊆ A. a) A ∪ ∅ b) A ∪ U c) A − U d) A ⊕ A e) ∅ − A f) A ∩ B g) A ∪ B h) A ∩ B i) A ⊕ B j) P(∅) (2) Using only p, q, r, ¬ and/or the connective ∧, write a proposition equivalent to each of the following (a) (p → q) → r (b) p → (q → r) (3) Write the contrapositive and converse of the statement: “You sleep late if it is Saturday”. (4) In the following, P(x, y) means “x + 2y = xy”. Where x and y are integers. Determine the truth value of the statement. (a) T F ∃yP(x, 3) (b) T F ∀x∃yP(x, y) (c) T F ∃x∀yP(x, y) (5) Suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course S(x): x is a sophomore F(x): x is a freshman T(x, y): x is taking y. Write the following statements using these predicates and any needed quantifiers. (a) There is a course that every freshman is taking. 1 (b) No freshman is a sophomore. (c) Some freshman is taking an advanced course. (d) There are at least two freshman students taking the exact same courses. (6) Determine whether the following argument is valid. p → r q → r ¬(p ∨ q) ∴ ¬r (7) Determine whether the following argument is valid. She is a Math Major or a Computer Science Major. If she does not know discrete math, she is not a Math Major. If she knows discrete math, she is smart. She is not a Computer Science Major. Therefore, she is smart. (8) Determine whether the rule describes a function. If your answer is no say why. (a) f : N → N where f(n) = √n. (b) g : N → N where g(n) = any integer > n. (9) Give an example of a function from Z to N that is both one-to-one and onto. (10) Give an example of a function from Z to N that is onto but NOT one-to-one. (11) Let f : A → B. Let B′ ⊂ B. Show that f(f −1(B′)) ⊆ B′. WHAT condition is needed for the containment in the other direction?