Name: Student number: METU MATH 111, Midterm 1 ^{1 2 3 4 5 Σ}

Thursday, November 3, 2011, at 17:40 (100 minutes), totally 60 points

Instructors: E.Emelyanov, G.Ercan, S.Finashin, E.Solak

Instructions: Please, show clearly the logic of your solutions.

Problem 1. (4+4+4 pts) (a) Is the statement (P ↔ Q)∧(P ∧¬Q) a tautology, a contradiction, or neither? (Explain.)

(b) Show that P → (Q ∨ R) is equivalent to (P ∧ ¬Q) → R.

(c) Record (using the logical symbols) the following claim. For every a, the equation ax^{2} + 4x − 2 = 0 has at least one solution iff a ≥ −2.

Problem 2. (3+3+3+3 pts) Suppose that P (x, y) means “x divides y”, and the universe is the set of positive integers. Determine whether the following statements are true or false. Explain (briefly).

(a) ∃y ∀x P (x, y)

(b) ∃x ∀y P (x, y)

(c) ∃x ∃y P (x, y)

(d) ∀x ∀y P (x, y)

Problem 3. (6+6 pts) Negate the following statements and simplify the result (the answer may contain negation only of P, Q, and R).

(a) (P ∨ ¬Q) ∧ ¬R

(b) ∀x (P (x) → Q(x))

Problem 4. (4+8 pts) (a) Suppose that A, B, and C are sets. Use appropriate logical symbols instead of set-operations “∩” to write that A ∩ B ∩ C ≠ ∅ (write an answer, do not prove).

(b) Suppose that X, Y, and Z are sets. Prove that X ∖ (Y ∖ Z) = (X ∖ Y) ∪ (X ∩ Z). Express clearly the logic!

Problem 5. (6+6 pts) Prove the following propositions. (Express clearly your logic).

(a) If n is an integer and n^{3} + 5 is odd, then n is even.

(b) There does not exist a positive real number a such that a + \frac{1}{a} < 2.