91Ó°ÊÓ

1\. Determine whether each of the following sentences is a statement. a) In 2003 George W. Bush was the president of the United States, b) \(x+3\) is a positive integer. c) Fifteen is an even number. d) If Jennifer is late for the party, then her cousin Zachary will be quite angry. e) What time is it? f) As of June 30,2003 , Christine Marie Evert had won the French Open a record seven times.

3\. Let \(p, q\) be primitive statements for which the implication \(p \rightarrow q\) is false. Determine the truth values for each of the following. a) \(p \wedge q\) b) \(\neg p \vee q\) c) \(q \rightarrow p\) d) \(\neg q \rightarrow \neg p\)

3\. Let \(p, q\), and \(r\) denote primitive statements. Prove or disorove (provide a counterexample for) each of the following. a) \([p \leftrightarrow(q \leftrightarrow r)] \Longleftrightarrow[(p \leftrightarrow q) \leftrightarrow r]\) b) \([p \rightarrow(q \rightarrow r)] \Longleftrightarrow[(p \rightarrow q) \rightarrow r]\)

4\. Let \(p, q, r, s\) denote the following statements: \(p\) : I finish writing my computer program before lunch. q: I shall play tennis in the afternoon. \(r\) : The sun is shining. s: The humidity is low. Write the following in symbolic form. a) If the sun is shining. I shall play tennis this afternoon. b) Finishing the writing of my computer program before lunch is necessary for my playing tennis this afternoon. c) Low humidity and sunshine are sufficient for me to play tennis this afternoon.

4\. Let \(n\) be a positive integer greater than 1 . We call \(n\) prime if the only positive integers that (exactly) divide \(n\) are 1 and \(n\) itself. For example, the first seven primes are \(2,3,5,7,11\), 13 , and 17. (We shall learn more about primes in Chapter 4.) Use the method of exhaustion to show that every integer in the universe \(4,6,8, \ldots, 36,38\) can be written as the sum of two primes.

4\. Consider the universe of all polygons with three or four ides, and define the following open statements for this universe. \(a(x)\) : all interior angles of \(x\) are equal \(e(x) ; \quad x\) is an equilateral triangle \(h(x)\) : all sides of \(x\) are equal \(i(x): x\) is an isosceles triangle \(p(x): \quad x\) has an interior angle that exceeds \(180^{\circ}\) \(q(x): \quad x\) is a quadrilateral \(r(x): \quad x\) is a rectangle \(s(x): \quad x\) is a square \(t(x): \quad x\) is a triangle Translate each of the following statements into an English sentence, and determine whether the statement is true or false. a) \(\forall x[q(x) \underline{\vee} t(x)]\) b) \(\forall x[i(x) \rightarrow e(x)]\) c) \(\exists x[t(x) \wedge p(x)]\) d) \(\forall x[(a(x) \wedge t(x)) \leftrightarrow e(x)]\) e) \(\exists x[q(x) \wedge \neg r(x)]\) f) \(\exists x[r(x) \wedge \neg s(x)]\) g) \(\forall x[h(x) \rightarrow e(x)]\) h) \(\forall x[t(x) \rightarrow \neg p(x)]\) i) \(\forall x[s(x) \leftrightarrow(a(x) \wedge h(x))]\) j) \(\forall x[t(x) \rightarrow(a(x) \leftrightarrow h(x))]\)

4\. For primitive statements \(p, q, r\), and \(s\), simplify the compound statement $$ [[[(p \wedge q) \wedge r] \vee[(p \wedge q) \wedge \neg r]] \vee \neg q] \rightarrow s $$

5\. Consider each of the following arguments. If the argument is valid, identify the rule of inference that establishes its validity. If not, indicate whether the error is due to an attempt to argue by the converse or by the inverse. a) Andrea can program in \(C++\), and she can program in Java. Therefore Andrea can program in \(\mathrm{C}++\). b) A sufficient condition for Bubbles to win the golf tournament is that her opponent Meg not sink a birdie on the last hole. Bubbles won the golf tournament. Therefore Bubbles' opponent Meg did not sink a birdic on the last hole. c) If Ron's computer program is correct, then he'll be able to complete his computer science assignment in at most two hours. It takes Ron over two hours to complete his computer science assignment. Therefore Ron's computer program is not correct. d) Eileen's car keys are in her purse, or they are on the kitchen table. Eileen's car keys are not on the kitchen table. Therefore Eileen's car keys are in her purse. e) If interest rates fall, then the stock market will rise. Interest rates are not falling. Therefore the stock market will not rise.

5\. Let \(p, q, r\) denote the following statements about a particular triangle \(A B C\). \(p\) : Triangle \(A B C\) is isosceles. q: Triangle \(A B C\) is equilateral. \(r:\) Triangle \(A B C\) is equiangular. Irite the following in symbolic form. a) If the sun is shining, I shall play tennis this afternoon. b) Finishing the writing of my computer program before lunch is necessary for my playing tennis this afternoon. c) Low humidity and sunshine are sufficient for me to play tennis this afternoon. 5\. Let \(p, q, r\) denote the following statements about a particlar triangle \(A B C .\) \(p\) : Triangle \(A B C\) is isosceles. q: Triangle \(A B C\) is equilateral. \(r\) : Triangle \(A B C\) is equiangular. ranslate each of the following into an English sentence. a) \(q \rightarrow p\) b) \(\neg p \rightarrow \neg q\) c) \(q \leftrightarrow r\) d) \(p \wedge \neg q\) e) \(r \rightarrow p\)

5\. Negate and express each of the following statements in smooth English. a) Kelsey will get a good education if she puts her studies before her interest in cheerleading. b) Norma is doing her homework, and Karen is practicing her piano lessons. c) If Harold passes his C++ course and finishes his data structures project, then he will graduate at the end of the semester.