2-2 Study Guide and Intervention

NAME ____________________________________________ DATE ____________________________ PERIOD _____________ 2-2 Study Guide and Intervention Statements, Conditionals, and Biconditionals Determine Truth Values A statement is any sentence that is either true or false. The truth value of a statement is either true (T) or false (F). A statement can be represented by using a letter. For example, Statement p: Chicago is a city in Illinois. The truth value of statement p is true. Several statements can be joined in a compound statement. Negation: not p is the negation of the statement p. Statement p and statement q joined by the word and is a conjunction. Statement p and statement q joined by the word or is a disjunction. Symbols: ~p (Read: not p) Symbols: p ∧ q (Read: p and q) Symbols: p ∨ q (Read: p or q) The statements p and ~p have opposite truth values. The conjunction p ∧ q is true only when both p and q are true. The disjunction p ∨ q is true if p is true, if q is true, or if both are true. Example 1: Write a compound statement for each Example 2: Write a compound statement for each conjunction. Then find its truth value. disjunction. Then find its truth value. p: An elephant is a mammal. p: A diameter of a circle is twice the radius. q: A square has four right angles. q: A rectangle has four equal sides. a. p ∧ q Join the statements with and: An elephant is a mammal and a square has four right angles. Both parts of the statement are true so the compound statement is true. a. p ∨ q Join the statements p and q with the word or: A diameter of a circle is twice the radius or a rectangle has four equal sides. The first part of the compound statement, p, is true, so the compound statement is true. b. ∽p ∧ q ∽p is the statement “An elephant is not a mammal.” Join ∽p and q with the word and: An elephant is not a mammal and a square has four right angles. The first part of the compound statement, ∽p, is false. Therefore the compound statement is false. b. ∽p ∨ q Join ∽p and q with the word or: A diameter of a circle is not twice the radius or a rectangle has four equal sides. Neither part of the disjunction is true, so the compound statement is false Exercises Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: 10 + 8 = 18 q: September has 30 days. r: A rectangle has four sides. 1. p and q 2. p ∨ r 3. q or r 4. q ∧ ∽r Chapter 2 12 Glencoe Geometry NAME ____________________________________________ DATE ____________________________ PERIOD _____________ 2-2 Study Guide and Intervention (continued) Statements, Conditionals, and Biconditionals Conditional Statements An if-then statement is a statement such as “If you are reading this page, then you are studying math.” A statement that can be written in if-then form is called a conditional statement. The phrase immediately following the word if is the hypothesis. The phrase immediately following the word then is the conclusion. A conditional statement can be represented in symbols as p → q, which is read “p implies q” or “if p, then q.” If you change the hypothesis or conclusion of a conditional statement, you form related conditionals. This chart shows the three related conditionals, converse, inverse, and contrapositive, and how they are related to a conditional statement. Symbols Formed by Example Conditional p→q using the given hypothesis and conclusion If two angles are vertical angles, then they are congruent. Converse q→p exchanging the hypothesis and conclusion If two angles are congruent, then they are vertical angles. Inverse ∽p → ∽q replacing the hypothesis with its negation and replacing the conclusion with its negation If two angles are not vertical angles, then they are not congruent. Contrapositive ∽q → ∽p negating the hypothesis, negating the conclusion, and switching them If two angles are not congruent, then they are not vertical angles. A conditional statement always has the same truth value as its contrapositive, and the converse and inverse always have the same truth value. If you make a compound statement from a conditional statement and its converse, you can write them as a single biconditional statement. Exercises Write each statement in if-then form. 1. All apes love bananas. 2. The sum of the measures of complementary angles is 90. 3. Collinear points lie on the same line. Write the converse, inverse, and contrapositive of each true conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. 4. If you live in San Diego, then you live in California. 5. If a polygon is a rectangle, then it is a square. Chapter 2 13 Glencoe Geometry