2.1 Homework 1) Label each of the following as a statement or as \"not a statement.\" If you find a statement, give its truth value. a) Take the highway. b) What time is it? c) Winston Churchill was the first president of the United States. 2) Consider the following two propositions: p: We can go to Cancun. q: We can go to Iceland. Using symbolic notation, a) Form the conjunction (). b) Form the disjunction (). c) Write the negation (~) of part a) as a logical statement and as an English sentence. d) Write the negation (~) of part b) as a logical statement and as an English sentence. 3) Consider the following statement: Every number is less than its square. For example, 5 < 25 and 7 < 49. Write the statement \"Every number is less that its square\" symbolically by defining a predicate and using a quantifier. 4) Let P(n): 2n = n a) What is P (2) as a statement? b) What is P (3) as a statement? c) Based off of this, would it be correct to say n P(n), n P(n), both, or neither? Explain. 5) Complete the following two truth tables. ( p p) p: P Q pVq (1) p (3) (2) ( p p) p: P T T F F q T F T F (p V~ q) (1) q (3) (2) Section 2.2 Homework 1) Use the following: p: It is raining. q: I stay home today. Write each of the following statement in terms of p and q using logical connectives. (Note: You may have to use negation, ~, in some instances): a) It is not raining implies that I stay home today. b) It is raining if I stay home today. c) I stay home today if and only if it is raining. d) It is raining only if I do not stay home today. e) It is raining is a sufficient condition for me to stay home today. f) It is raining is a necessary condition for me to stay home today 2) Write the contrapositive and the converse of the following statement: If the post office is open, then it is not a national holiday. a) Contrapositive: b) Converse: 3) Complete the truth table for P v qp P v q p Explain how the truth table shows us that P v q p is a tautology, a contingency, or an absurdity. Also explain how the truth table allows you to descredit the other two possibilities. Section 2.3 Homework For problems 1-3, state whether the argument given is valid or not. Write the argument symbolically. For example, of the form (pq)~q~p. 1) If the TV is on sale, I will buy the TV. The TV is not on sale. I will not buy the TV. 2) If the TV is on sale or I can finance it, I will buy the TV. I cannot finance the TV. I will not buy the TV. 3) The TV is not on sale or I will buy it. The TV is on sale. I will buy the TV. 4) Prove that if n is odd, then n is odd 5) Consider the fact (0.5) = 0.25 and 0.25< 0.5 Prove or disprove: For any real number, x < x