Concept 8 Inductive Reasoning

A Conjecture is an unproven statement that is based on observations. Example 1: Sketch the next figure in the pattern.

You use Inductive reasoning when you find a pattern in specific cases and write a conjecture for the general case. Example 2: Describe a pattern in the numbers. Write the next number in the pattern. 8, 0, -8, -16, -24, … 1, 4, 9, 16, 25, … Example 3: The first three objects in a pattern are shown. How many squares are in the next figure? a) b) Subtract 8 from the previous number. So – 24 – 8 = – 32 The next perfect square Ex. 1 2 , 2 2 , 3 2 , 4 2 , 5 2 , so 6 2 = 36

a specific case for which a conjecture is false. (an example) A Counterexample is a specific case for which a conjecture is false. (an example) Example 4: Show the conjecture is false by finding a counterexample. Every even number is composite. The square root of a number x is always less than x. Composite means can be divided by more than 1 and itself. 2 is an even number by it is not composite it is a prime number. 1 4 < 1 4 𝑥 <𝑥 So try some: 4 <4 2 < 4 true 1 2 < 1 4 false

Compound Statements and Truth Tables Concept 9

Statement: A sentence that is a complete though that states a given fact, either true or false. Example: p: It is raining outside. q: A right angle does not measure 90 degrees. r: 2 + 8 = 10 Negation: The opposite of a statement, usually found by adding or taking away the word(s) not, never, none, not equal to, … Example: ~p: It is not raining outside. ~q: A right angle measures 90 degrees. ~r: 2 + 8 ≠ 10 Read “not p”

∧ ∨ Compound Statement: A statement made by combing two statements using the words “and” or “or”. p: A rectangle is a quadrilateral. q: A rectangle is convex. A) Conjunction: B) Disjunction: To combine with the word “and”. To combine with the word “or”. ∧ ∨ Symbol: Symbol: Examples: 1) p ∧ q 2) ~p ∧ q A rectangle is a quadrilateral and a rectangle is convex. A rectangle is a quadrilateral or a rectangle is convex. 1) p ∨ q 2) p ∨ ~q A rectangle is not a quadrilateral and a rectangle is convex. A rectangle is a quadrilateral or a rectangle is not convex.

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   3. q or r    2. p ∨ r 10 + 8 = 18 or a rectangle has four sides. 10 + 8 = 18 and September has 30 days. 4. q ∧ ∽r September has 30 days and a rectangle does not have four sides. September has 30 days or a rectangle has four sides.

Negation Conjunction Disjunction Truth Table- a convenient way to organize the possible outcomes of a statement. With “and” both or all have to be true for the outcome to be true. With “or” at least one has to be true for the outcome to be true. Negation P ~P   Conjunction p q p ˄ q   Disjunction p q p ˅ q   T F T T T T T T F T T F F T F T F T F F T T F F F F F F What are the possibilities for p? What are the opposite possibilities for p? Always alternate true then false. Always 2 true then 2 false.

or and Examples 1) ~p ˅ q 2) p ˄ ~q p q ~p ~p ˅ q p q ~q p ˄ ~q T T F   p q ~p ~p ˅ q   p q ~q p ˄ ~q T T F T T T F F T F F F T F T T F T T T F T F F F T F T F F T F

Examples 3) ~p ˄ ~q 4) p ˄ ( ~q ˅ r) p q ~p ~q ~p ˄ ~q p q r ~q ~q ˅ r   p q ~p ~q ~p ˄ ~q   p q r ~q ~q ˅ r p ˄ ( ~q ˅ r) T T F F F T T T F T T T F F T F T T F F F F F T T F F T F T T T T F F T T T T F F T T T F T T F T F F T F F F F F F T T T F F F F T T F

Construct a truth table for each compound statement.  1. p or r 2. ∽p ∨ q   p q p or r   p q ~p ~ p or r T T T T T F T T T F T F F F F T T F T T T F T F F F F T

3. p ∨ (q ∧ ∽r) p q r ~r q ˄ ~r p ˅ ( q ˄ ~r) T T T F F T T T F T T T   p q r ~r q ˄ ~r p ˅ ( q ˄ ~r) T T T F F T T T F T T T T F T F F T T F F T F T F T T F F F F T F T T T F F T F F F F F F T F F

4. (p ∧ r) ∨ q p q r p ˄ r ( p ˄ r) ˅ q T T T T T T T F F T T F T T T   p q r p ˄ r ( p ˄ r) ˅ q T T T T T T T F F T T F T T T T F F F F F T T F T F T F F T F F T F F F F F F F

A way to organize data about subjects that have things in common. Venn diagram - A way to organize data about subjects that have things in common. p: A rectangle is a quadrilateral. q: A rectangle is convex. ~p ∧ ~q p q ~p ˄ q p ˄ q p ˄ q p ˄ ~q ~p p ˅ q ~q ~p ∨ ~q

25 people attended meetings in May or June, of those 5 attended only in May, 14 attended only in June and some attended both months. There were two people that didn’t attend any meetings at all. Make a Venn diagram to find the number of people that attended both the May and June meetings. June May 5 6 14 2 25

The Venn diagram shows the number of graduates last year who did or did not attend their junior or senior prom. A. How many graduates attended their senior but not their junior prom? Which part is: senior ∧ not junior 25 B. How many graduates attended their junior and senior proms? Which part is: junior ∧ junior 123 C. How many graduates did not attend either of their proms? Which part is: not senior ∧ not junior 37 D. How many students graduated last year? Explain. There were 123 + 25 seniors so 148 graduated. Junior 85 123 37 Senior 25

A group of students were surveyed on the type of pet they owned A group of students were surveyed on the type of pet they owned. The Venn diagram represents the data found. How many students have a cat and some other animal? How many students have a dog? How many only have a dog, cat, and other animal? How many students only have a dog? 10 + 4 = 14 15 + 5 + 10 + 8 = 38 10 15

How many students have a dog and other animal but not a cat? How many students have a dog and cat? How many have a dog, cat, or other animal? How many students have a cat or dog, but not any other animal? 5 + 10 = 15 but 15 – 10 = 5 10 + 8 = 18 15 + 5 + 10 + 8 + 18 + 4 + 6 = 66 15 + 5 + 10 + 8 + 18 + 4 = 60 But 60 – 5 – 4 – 10 = 41 for not other animal.

Conditional Statements Concept 10

Conditional Statement - A logical statement that has two parts; a hypothesis and a conclusion joined with the words “if” and “then”. Conditional Statement - Hypothesis - the “if” part of the sentence, or what has to happen first Conclusion - the “if” part of the sentence, or what has to happen first p q Hypothesis Conclusion If p then q. Or p  q Notice that the words “if” and “then” are not a part of the hypothesis or conclusion.

Write an equivalent conditional statement in if-then form. If-Then Statements Example 1: Write an equivalent conditional statement in if-then form. A. A right angle is an angle that measures exactly 90°. I bring my umbrella when it is raining. Suzanne eats a snack if she is hungry. Hypotenuse Another word for “then” This would be the conclusion. If an angle is a right angle, then it measures exactly 𝟗𝟎°. Conclusion Another word for “if” Hypotenuse If it is raining, then I bring my umbrella. Conclusion Hypotenuse Here is an “if” If Suzanne is hungry, then she eats a snack.

Other types of Conditional Statements. Negation: The opposite of a statement, usually found by adding or taking away the word(s) not, never, none, not equal to, … Converse: To switch the hypothesis and conclusion of a conditional statement. Conditional Statement: q  p Inverse: The negation of the conditional statement. p  q ~p  ~q Contrapositive: The negation of the converse. ~q  ~p Note: A conditional statement and its _______________ are either both true or both false. Also, the converse and ____________ are either both true or both false. contrapositive inverse

Converse, Inverse, & Contrapositive Practice Example 2: Write the converse, inverse, and contrapositive of each statement. Decide whether each statement is true or false. If a polygon has three sides, then it is a triangle. If you play football, then you are an athlete. T Converse: If a polygon is a triangle, then it has three sides. T Inverse: If a polygon does not have three sides, then it is not a triangle. T T Contrapositive: If a polygon is not a triangle, then it does not have three sides. T Converse: If you are an athlete, then you play football. F Inverse: If you do not play football, then you are not an athlete. F Contrapositive: If you are not an athlete, then you do not play football. T

Biconditional Statements A statement that combines a true conditional statement with its true converse using the words “if and only if”. _________________ can always be written as biconditional statements. Good definitions Example 3: Write the converse of each true if-then statement. If the converse is also true, combine the statements to write a true biconditional statement. If an angle measures greater than 0° and less than 90°, then it is acute. T If an angle is acute, then it measures greater than 0° and less than 90°. T An angle measures greater than 𝟎° and less than 𝟗𝟎° if and only if it is acute.

B. If two angles are complementary, then the sum of their measures is 90°. C. If a number is an integer, then it is rational. T If the sum of two angles measures is 90°, then they are complementary angle. T Two angles are complementary if and only if the sum of their measures is 𝟗𝟎°. T If a number is rational, then it is an integer. F

Example 4: Write the biconditional statement as a conditional statement and its converse. Two angles are adjacent if and only if they share a common vertex and a common side. Two angles are complementary if and only if the sum of their measures is 90°. Conditional: If two angles are adjacent, then they share a common vertex and a common side. Converse: If two angles share a common vertex and a common side, then they are adjacent. Conditional: If two angles are complementary, then the sum of their measures is 90°. Converse: If the sum of two angle measures is 90°, then they are complementary.

C. I will graduate from high school if and only if I finish all the requirements. Conditional: If I will graduate from high school then I finished all the requirements. Converse: If I finished all the requirements, then I will graduate from high school.

Deductive reasoning Concept 11 Uses facts, definitions, accepted properties, and laws of logic to form a logical argument.

Law of Detatchment If the hypothesis of a true conditional statement is true, then the conclusion is also true. Use the Law of Detachment to make a valid conclusion in the true statement. 1. If it is raining, then the soccer game will be canceled. It is raining. The soccer game will be canceled. 2. If I study for my Geometry exam, then I will get a good grade. I studied for my Geometry exam. I will get a good grade. 3. If the measure of an angle is 90°, then it is a right angle. The measure of angle A is 90°. Angle A is a right angle.

Law of Syllogism If these statements are true, If hypothesis p, then conclusion q and If hypothesis q, then conclusion r. Then the following statement is true. If hypothesis p, then conclusion r. If there exists pq and qr that are true, then pr is also a true statement. If possible, use of the Law of Syllogism to write the conditional statement that follows from the pair of true statements. 1. If it continues to rain, then the field will get wet. If the field is wet, then the soccer game will be cancelled. If it continues to rain, then the soccer game will be canceled.

2. If you study hard, you will pass all of your classes 2. If you study hard, you will pass all of your classes. If you pass all of your classes, then you will graduate from high school. If you study hard, then you will graduate from high school. 3. If a triangle has two angles that each measure 45°, then the third angle measures 90°. If an angle in a triangle measures 90°, then it is a right triangle. If a triangle has two angles that each measure 45°, then it is a right triangle.

Determine if the conclusion reached from the two statements demonstrates the Law of Detachment, the Law of Syllogism, or neither. 1. If the electric power is off, then the refrigerator does not work. If the refrigerator does not work, then the food will spoil. If the electric power is off, then the food will spoil. Law of Syllogism 2. If you order a quesadilla, then it will be served with guacamole. Chloe ordered a quesadilla. Chloe was served guacamole. Law of Detachment 3. If two integers are added together, then their sum is an integer. Integer X is added to integer Y. The sum of X and Y is an integer. Law of Detachment

4. If you play the tuba, then you play a brass instrument 4. If you play the tuba, then you play a brass instrument. Joshua plays a brass instrument. Joshua plays the tuba. Neither 5. If I miss my bus, then I will be late for school. If I’m late to school, then I will get a detention. If I miss my bus, then I will get a detention. Law of Syllogism