AMSCO'S Geometry. New York - Rye High School

90 LogicExploration1. A tautology is a statement that is always true. For instance, the disjunctionp ∨ p is a tautology because if p is true, the disjunction is true, and if p isfalse, p is true and the disjunction is true.a. Which of the following statements are tautologies?(1) p → (p ∨ q)(2) Either it will rain or it will snow.(3) (p ∧ q) → pb. Construct two tautologies, one using symbols and the other using words.2. A contradiction is a statement that is always false, that is, it cannot be trueunder any circumstances. For instance, the conjunction p ∧ p is a contradictionbecause if p is true, p is false and the conjunction is false, and if pis false, the conjunction is false.a. Which of the following statements are contradictions?(1) (p ∨ q) ∧ p(2) It is March or it is Tuesday, and it is not March and it is not Tuesday.(3) (p ∧ p) ∧ qb. Construct two contradictions, one using symbols and the other using words.3. One way to construct a tautology is to use a contradiction as the hypothesisof a conditional. For instance, since we know that p ∧ p is a contradiction,the conditional (p ∧ p) → q is a tautology for any conclusion q.a. Construct two conditionals using any two contradictions as thepremises.b. Show that the two conditionals from part a are tautologies.c. Explain why the method of using a contradiction as the hypothesis of aconditional always results in a tautology.CUMULATIVE REVIEW CHAPTERS 1–2Part IAnswer all questions in this part. Each correct answer will receive 2 credits. Nopartial credit will be allowed.1. Which of the following is an undefined term?(1) ray (2) angle (3) line (4) line segment2. The statement “If x is a prime then x is odd” is false when x equals(1) 1 (2) 2 (3) 3 (4) 4