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Additional Topics for the Mid-Term Exam Propositions • Definition: A proposition is a statement with a truth value. That is, it is a statement that is true or else a statement that is false. • Next are some examples with their truth values. Propositions • Denver is the capital of Colorado. (true) • The main campus of the University of Kentucky is in Athens, Ohio. (false) • It will rain in Knoxville tomorrow. (We cannot know the truth of this statement today, but it is certainly either true or false.) • 2+2=4 (true) • 2+2=19 (false) • There are only finitely many prime numbers. (false) Propositions • On the other hand, here are some examples of expressions that are not propositions. Not Propositions • Where is Denver? (This is neither true nor false. It is a question.) • Find Denver! (This is neither true nor false. It is a command.) • Blue is the best color to paint a house. (This is a matter of opinion, not truth. It may be your favorite color, but it is not objective truth.) • Not Propositions – Questions – Command – Opinion Propositions • Notation: represented by a lower case letter, usually p, q & r. • Example: designating a proposition; p: the glass is full. q: I am 35 years old. r: today is Monday. Negation • Negation (not): The negation of a proposition p is “not p”. • We denote it ~p. • It is true if p is false and vice versa. 7 Negation • Formed by inserting “not” or “do not” in the proposition. • Example; p: Harvard is a college. ~p: Harvard is not a college. q: I am 35 years old. ~q: I am not 35 years old. Conditional • A conditional is a compound proposition of the form “if p then q”. • We denote the compound proposition “if p then q” by p → q. • The conditional “if p then q” or “p→q” is made up of two parts; 1. Hypothesis – proposition p in the above conditional. 2. Conclusion – proposition q in the above conditional. 9 Example Conditional 1. Given p: John is not at work and q: John is sick. Find p→q. p→q: If John is not at work then John is sick. 2. Given the conditional, If a number is divisible by four then it is even. Identify the hypothesis and the conclusion. – Hypothesis: A number is divisible by four. – Conclusion: A number is even. If-Then Statements • A conditional statement is a statement that can be written in if-then form. Example: If an animal has hair, then it is a mammal. • Conditional statements are always written “if p, then q.” The phrase which follows the “if” (p) is called the hypothesis, and the phrase after the “then” (q) is the conclusion. • We write p → q, which is read “if p, then q”. Example 1: Identify the hypothesis and conclusion of the following statement. If a polygon has 6 sides, then it is a hexagon. If a polygon has 6 sides, then it is a hexagon. hypothesis conclusion Answer: Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon Your Turn: Identify the hypothesis and conclusion of each statement. a. If you are a baby, then you will cry. Answer: Hypothesis: you are a baby Conclusion: you will cry b. To find the distance between two points, you can use the Distance Formula. Answer: Hypothesis: you want to find the distance between two points Conclusion: you can use the Distance Formula If-Then Statements • Often, conditionals are not written in “if-then” form but in standard form. By identifying the hypothesis and conclusion of a statement, we can translate the statement to “if-then” form for a better understanding. • When writing a statement in “if-then” form, identify the requirement (condition) to find your hypothesis and the result as your conclusion. Example 2: Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. Distance is positive. Sometimes you must add information to a statement. Here you know that distance is measured or determined. Answer: Hypothesis: a distance is determined Conclusion: it is positive If a distance is determined, then it is positive. Your Turn: Identify the hypothesis and conclusion of each statement. Then write each statement in if-then form. a. A polygon with 8 sides is an octagon. Answer: Hypothesis: a polygon has 8 sides Conclusion: it is an octagon If a polygon has 8 sides, then it is an octagon. b. An angle that measures 45º is an acute angle. Answer: Hypothesis: an angle measures 45º Conclusion: it is an acute angle If an angle measures 45º, then it is an acute angle. Converse, Inverse, and Contrapositive • From a conditional we can also create additional statements referred to as related conditionals. These include the converse, the inverse, and the contrapositive. Converse, Inverse, and Contrapositive • Writing the Converse, Inverse & Contrapositive of an Conditional • Given the conditional p→q – Converse is q→p (reverse the hypothesis and the conclusion). – Inverse is ~p→~q (negate both the hypothesis and the conclusion). – Contrapositive is ~q→~p (reverse and negate the hypothesis and the conclusion). Converse, Inverse, and Contrapositive Example • p: a quadrilateral is a rectangle q: a quadrilateral is a parallelogram p→q: if a quadrilateral is a rectangle then a quadrilateral is a parallelogram Find the converse, inverse and contrapositive in both logic notation and in words. 20 Solution • Converse; q→p: if a quadrilateral is a parallelogram then a quadrilateral is a rectangle. • Inverse; ~p→~q: if a quadrilateral is not a rectangle the a quadrilateral is not a parallelogram. • Contrapositive; ~q→~p: if a quadrilateral is not a parallelogram then a quadrilateral is not a rectangle. 21 Truth Value of Contrapositive • Law of Contrapositives: If an conditional is true, then it’s contraposive is also true and conversely (ie. The contrapositive always has the same truth value as the original conditional). Converse, Inverse, and Contrapositive • Statements that have the same truth value are said to be logically equivalent. We can create a truth table to compare the related conditionals and their relationships. Conditional Converse Inverse Contrapositive p q p→q q→p ~p →~q ~q →~p T T T T T T T F F T T F F T T F F T F F T T T T Example 4: Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. First, write the conditional in if-then form. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with 𝓁 = 2 and w = 4 is not a square. Example 4: Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square, but it is a rectangle. The contrapositive is the negation of the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true. Your Turn: Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90. Determine whether each statement is true or false. If a statement is false, give a counterexample. Answer: Conditional: If two angles are complementary, then the sum of their measures is 90; true. Converse: If the sum of the measures of two angles is 90, then they are complementary; true. Inverse: If two angles are not complementary, then the sum of their measures is not 90; true. Contrapositive: If the sum of the measures of two angles is not 90, then they are not complementary; true. Kites What makes a quadrilateral a kite? Definition: Kite A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent or parallel. Angles of a Kite You can construct a kite by joining two different isosceles triangles with a common base and then by removing that common base. Two isosceles triangles can form one kite. Angles of a Kite • Just as in an isosceles triangle, the angles between each pair of congruent sides are vertex angles. The other pair of angles are nonvertex angles. Kite Theorem 6.25 • If a quadrilateral is a kite, then its diagonals are perpendicular. Kite Theorem 6.26 • If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. Example 7A A. If WXYZ is a kite, find mXYZ. Example 7A Since a kite only has one pair of congruent angles, which are between the two non-congruent sides, WXY WZY. So, WZY = 121. mW + mX + mY + mZ = 360 73 + 121 + mY + 121 = 360 mY = 45 Answer: mY = 45 Polygon Interior Angles Sum Theorem Substitution Simplify. Example 7B B. If MNPQ is a kite, find NP. Example 7B Since the diagonals of a kite are perpendicular, they divide MNPQ into four right triangles. Use the Pythagorean Theorem to find MN, the length of the hypotenuse of right ΔMNR. NR2 + MR2 = MN2 (6)2 + (8)2 = MN2 36 + 64 = MN2 100 = MN2 10 = MN Pythagorean Theorem Substitution Simplify. Add. Take the square root of each side. Example 7B Since MN NP, MN = NP. By substitution, NP = 10. Answer: NP = 10 Your Turn: A. If BCDE is a kite, find mCDE. A. 28° B. 36° C. 42° D. 55° Your Turn: B. If JKLM is a kite, find KL. A. 5 B. 6 C. 7 D. 8 6.25 6.26