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Represent each of the following statements by a formula. What is their truth value if r is true? What if r is false? 28 Chapter 2 Logic (a) Niagara Falls is in New York if and only if New York City is the state capital of New York. (b) Niagara Falls is in New York iff New York City will have more than 40 inches of snow in 2525. (c) Niagara Falls is in New York or New York City is the state capital of New York if and only if New York City will have more than 40 inches of snow in 2525. 4. Express each of the following compound statements symbolically: (a) (b) (c) (d) (e) (f) The product xy = 0 if and only if either x = 0 or y = 0.

D) If eπ is a real number, then eπ is either rational or irrational. (e) All NFL players are huge. 6. Find the converse, inverse, and contrapositive of the following implication: If the quadrilateral ABCD is a rectangle, then ABCD is a parallelogram. 7. Construct the truth tables for the following expressions: (a) (p ∧ q) ∨ r (b) (p ∨ q) ⇒ (p ∧ r) Hint: To help you get started, fill in the blanks. (a) p T T T T F F F F r p ∧ q (p ∧ q) ∨ r T F T F T F T F q T T F F T T F F (b) p T T T T F F F F q T T F F T T F F r T F T F T F T F p∨q p∧r (p ∨ q) ⇒ (p ∧ r) 8.

Hence, f (c) = 0 for some number c between a and b. 5 The function f (x) = 5x3 − 2x − 1 is a polynomial function, which is known to be continuous over the real numbers. 3 implies that there exists a number between 0 and 1 such that 5x3 − 2x − 1 = 0. 6 between 0 and π2 . Show that the equation 1+x cos x = 0 has at least one real solution Hint: No function is mentioned here, so you need to define a function, say g(x). Next, you need to make sure that g(x) is continuous. 3? Summary and Review • Sometimes we can prove a statement by showing how the result can be obtained through a construction, and we can describe the construction in an algorithm.

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