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”.

Identify the hypothesis and conclusion of the following statement. If a polygon has 6 sides, then it is a hexagon. Answer: Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon hypothesis conclusion If a polygon has 6 sides, then it is a hexagon. Example 1:

Identify the hypothesis and conclusion of each statement. a. If you are a baby, then you will cry. b. To find the distance between two points, you can use the Distance Formula. Answer: Hypothesis: you are a baby Conclusion: you will cry Answer: Hypothesis: you want to find the distance between two points Conclusion: you can use the Distance Formula Your Turn:

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.

Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. Answer: Hypothesis: a distance is determined Conclusion: it is positive If a distance is determined, then it is positive. Sometimes you must add information to a statement. Here you know that distance is measured or determined. Distance is positive. Example 2:

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. b. An angle that measures 45º is an acute angle. Answer: Hypothesis: a polygon has 8 sides Conclusion: it is an octagon If a polygon has 8 sides, then it is an octagon. Answer: Hypothesis: an angle measures 45º Conclusion: it is an acute angle If an angle measures 45º, then it is an acute angle. Your Turn:

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. ConditionalConverseInverseContrapositive pq p → q q → p ~p →~q ~q →~p TTTTTT TFFTTF FTTFFT FFTTTT

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. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. First, write the conditional in if-then form. 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 = 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. Example 4:

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. Your Turn:

Kites What makes a quadrilateral a kite?

Definition: Kite 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 vertex angles nonvertex angles 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 m  XYZ.

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 . m  W + m  X + m  Y + m  Z=360Polygon Interior Angles Sum Theorem m  Y + 121=360Substitution m  Y=45Simplify. Answer: m  Y = 45

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. NR 2 + MR 2 =MN 2 Pythagorean Theorem (6) 2 + (8) 2 =MN 2 Substitution =MN 2 Simplify. 100=MN 2 Add. 10=MNTake the square root of each side.

Example 7B Answer: NP = 10 Since MN  NP, MN = NP. By substitution, NP = 10.

Your Turn: A.28° B.36° C.42° D.55° A. If BCDE is a kite, find m  CDE.

Your Turn: A.5 B.6 C.7 D.8 B. If JKLM is a kite, find KL.