2-3: deductive reasoning

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03/12/22 03/12/22 2-3: Deductive Reasoning 2-3: Deductive Reasoning 1 1 2-3: Deductive 2-3: Deductive Reasoning Reasoning Expectations: Expectations: L3.1.1: Distinguish between inductive and L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing deductive reasoning, identifying and providing examples of each. examples of each. L3.1.3: Define and explain the roles of axioms L3.1.3: Define and explain the roles of axioms (postulates), definitions, theorems, (postulates), definitions, theorems, counterexamples, and proofs in the logical counterexamples, and proofs in the logical structure of mathematics. Identify and give structure of mathematics. Identify and give examples of each. examples of each. L3.3.3: Explain the difference between a L3.3.3: Explain the difference between a necessary and a sufficient condition within the necessary and a sufficient condition within the statement of a theorem. Determine the correct statement of a theorem. Determine the correct conclusions based on interpreting a theorem in conclusions based on interpreting a theorem in which necessary or sufficient conditions in the which necessary or sufficient conditions in the theorem or hypothesis are satisfied. theorem or hypothesis are satisfied.

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2-3: Deductive Reasoning. Expectations: L3.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each. - PowerPoint PPT Presentation

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04/19/2304/19/23 2-3: Deductive Reasoning2-3: Deductive Reasoning 11

2-3: Deductive 2-3: Deductive ReasoningReasoning

Expectations:Expectations:L3.1.1: Distinguish between inductive and deductive L3.1.1: Distinguish between inductive and deductive

reasoning, identifying and providing examples of each.reasoning, identifying and providing examples of each.L3.1.3: Define and explain the roles of axioms (postulates), L3.1.3: Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the definitions, theorems, counterexamples, and proofs in the

logical structure of mathematics. Identify and give examples logical structure of mathematics. Identify and give examples of each.of each.

L3.3.3: Explain the difference between a necessary and a L3.3.3: Explain the difference between a necessary and a sufficient condition within the statement of a theorem. sufficient condition within the statement of a theorem.

Determine the correct conclusions based on interpreting a Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem in which necessary or sufficient conditions in the

theorem or hypothesis are satisfied.theorem or hypothesis are satisfied.

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DiagonalsDiagonals

A segment is a diagonal of a polygon iff its A segment is a diagonal of a polygon iff its endpoints are 2 non-consecutive vertices endpoints are 2 non-consecutive vertices of a polygon.of a polygon.

ex: AC, BE and DF are diagonals for ex: AC, BE and DF are diagonals for polygon ABCDEF.polygon ABCDEF.

B

C D

E

FA

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Use a pattern to answer the Use a pattern to answer the question.question.

How many diagonals does an octagon How many diagonals does an octagon have? have?

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Patterns are not proof – they are conjecture. Patterns are not proof – they are conjecture. Remember, this is inductive reasoning Remember, this is inductive reasoning which is not valid for making a proof.which is not valid for making a proof.

The following slides give us some properties The following slides give us some properties of deductive reasoning which is valid for of deductive reasoning which is valid for proving statements true.proving statements true.

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Law of Detachment Law of Detachment

If p => q is true and p is a true statement, If p => q is true and p is a true statement, then ___ must be true.then ___ must be true.

ex:ex:

1. If today is Monday, then tomorrow is 1. If today is Monday, then tomorrow is Tuesday.Tuesday.

2. Today is Monday.2. Today is Monday.

Conclude: _________________.Conclude: _________________.

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Make a conclusion based on the following Make a conclusion based on the following true statements.true statements.

a. If the air conditioner is on, then it is hot a. If the air conditioner is on, then it is hot outside.outside.

b. The air conditioner is on. b. The air conditioner is on.

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Make a conclusion based on the following Make a conclusion based on the following true statements.true statements.

a. If it is raining, then it is humid.a. If it is raining, then it is humid.

b. It is humid. b. It is humid.

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If p => q and q are true If p => q and q are true _________________ can be made. _________________ can be made.

This is referred to as affirming the This is referred to as affirming the consequent.consequent.

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Necessary and Sufficient Necessary and Sufficient ConditionsConditions

In the statement of a theorem in “if- then” In the statement of a theorem in “if- then” form, we can talk about sufficient form, we can talk about sufficient conditions for the truth of the statement conditions for the truth of the statement and necessary conditions of the truth of and necessary conditions of the truth of the statement.the statement.

This is really just another way of looking at This is really just another way of looking at the Law of Detachment and Affirming the the Law of Detachment and Affirming the Consequent.Consequent.

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The ___________ is a sufficient condition The ___________ is a sufficient condition for the conclusion and the conclusion is a for the conclusion and the conclusion is a _____________ condition of the _____________ condition of the hypothesis.hypothesis.

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NecessaryNecessary

Consider the statement p => q. We say q is Consider the statement p => q. We say q is a necessary condition for (or of) p. a necessary condition for (or of) p.

Ex: “If if is Sunday, then we do not have Ex: “If if is Sunday, then we do not have school.”school.”

A necessary condition of it being Sunday is A necessary condition of it being Sunday is that we do not have school.that we do not have school.

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Sufficient ConditionSufficient Condition

A sufficient condition is a condition that all A sufficient condition is a condition that all by itself guarantees another statement by itself guarantees another statement must be true.must be true.

Ex: If you legally drive a car, then you are at Ex: If you legally drive a car, then you are at least 15 years old.”least 15 years old.”

Driving legally guarantees that a person Driving legally guarantees that a person must be at least 15 years old.must be at least 15 years old.

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Notice that, “We do not have school today” Notice that, “We do not have school today” is not sufficient to guarantee that today is is not sufficient to guarantee that today is Sunday.Sunday.

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““If M is the midpoint of segment AB, If M is the midpoint of segment AB, then AM then AM ≅≅ MB.” MB.”

Given that M is the midpoint, it is necessary Given that M is the midpoint, it is necessary (true) that AM (true) that AM ≅≅ MB. MB.

This means that M being the midpoint is a This means that M being the midpoint is a ____________ condition for AM ____________ condition for AM MB. MB.

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Notice simply saying AM Notice simply saying AM ≅≅ MB does not MB does not guarantee that M is the midpoint of AB, so guarantee that M is the midpoint of AB, so it is not a sufficient condition.it is not a sufficient condition.

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““If a triangle is equilateral, then it is If a triangle is equilateral, then it is isosceles.”isosceles.”

A triangle having 3 congruent sides A triangle having 3 congruent sides (equilateral) guarantees that at least 2 (equilateral) guarantees that at least 2 sides are congruent, so a triangle being sides are congruent, so a triangle being equilateral is sufficient to say it is equilateral is sufficient to say it is isosceles.isosceles.

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““If a person teaches mathematics, If a person teaches mathematics, then they are good at algebra.”then they are good at algebra.”

Because Trevor is a math teacher, can we Because Trevor is a math teacher, can we conclude he is good at algebra. Justify your conclude he is good at algebra. Justify your answer.answer.

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““If a person teaches mathematics, If a person teaches mathematics, then they are good at algebra.”then they are good at algebra.”

Betty is 32 and is very good at algebra. Can we Betty is 32 and is very good at algebra. Can we correctly conclude that she is a math teacher? correctly conclude that she is a math teacher? Justify.Justify.

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Bi-Conditional StatementsBi-Conditional Statements

If a statement and its converse are both true If a statement and its converse are both true it is called a bi-conditional statement and it is called a bi-conditional statement and can be written in ________________ can be written in ________________ form.form.

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Ex:Ex:““If an angle is a right angle, then its measure If an angle is a right angle, then its measure

is exactly 90is exactly 90°” and “If the measure of an °” and “If the measure of an angle is exactly 90°, then it is a right angle is exactly 90°, then it is a right angle” are true converses of each other so angle” are true converses of each other so they can be combined into a single they can be combined into a single statement.statement.

________________________________________________________________________________________________________________________________________

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Necessary Necessary andand Sufficient Sufficient

If a statement is a bi-conditional statement If a statement is a bi-conditional statement then either part is a necessary and then either part is a necessary and sufficient condition for the entire sufficient condition for the entire statement.statement.

Remember all definitions are bi-conditional Remember all definitions are bi-conditional statements.statements.

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A triangle is a right triangle iff it has a right A triangle is a right triangle iff it has a right angle.angle.

Being a right triangle is necessary and Being a right triangle is necessary and sufficient for a triangle to have a right sufficient for a triangle to have a right angle and possessing a right angle is angle and possessing a right angle is necessary and sufficient for a triangle to necessary and sufficient for a triangle to be a right triangle.be a right triangle.

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Necessary, Sufficient, Both or Necessary, Sufficient, Both or NeitherNeither

Given the true statement:Given the true statement:

““If a quadrilateral is a rhombus, then its If a quadrilateral is a rhombus, then its diagonals are perpendicular.”diagonals are perpendicular.”

Is the following statement necessary, Is the following statement necessary, sufficient, both or neither?sufficient, both or neither?

The diagonals of ABCD are perpendicular.The diagonals of ABCD are perpendicular.

Which of the following is a sufficient but NOT Which of the following is a sufficient but NOT necessary condition for angles to be necessary condition for angles to be supplementary?supplementary?

A.A. they are both acute angles.they are both acute angles.

B.B. they are adjacentthey are adjacent

C.C. their measures add to 90.their measures add to 90.

D.D. they are coplanar.they are coplanar.

E.E. they form a linear pair.they form a linear pair.

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Necessary, Sufficient, Both or Necessary, Sufficient, Both or NeitherNeither

Given the true statement:Given the true statement:

““A quadrilateral is a rhombus if and only if its A quadrilateral is a rhombus if and only if its 4 sides are congruent.”4 sides are congruent.”

Is the following statement necessary, Is the following statement necessary, sufficient, both or neither?sufficient, both or neither?

The sides of ABCD are all congruent.The sides of ABCD are all congruent.

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Law of Syllogism: Transitive Law of Law of Syllogism: Transitive Law of Logic (A Form of Logical Argument)Logic (A Form of Logical Argument)

If p => q and q =>r, then __________. If p => q and q =>r, then __________.

ex:ex:

1. If a polygon is a square, then it is a 1. If a polygon is a square, then it is a rhombus.rhombus.

2. If a polygon is a rhombus, then it is a 2. If a polygon is a rhombus, then it is a parallelogram.parallelogram.

Conclude: __________________________Conclude: __________________________

__________________________________.__________________________________.

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Make a conclusion based on the following.Make a conclusion based on the following.

a. If a quadrilateral is a square, then it has a. If a quadrilateral is a square, then it has 4 right angles.4 right angles.

b. If a quadrilateral has 4 right angles, then b. If a quadrilateral has 4 right angles, then it is a rectangle.it is a rectangle.

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Assignment Assignment

pages 89 - 91,pages 89 - 91,

# 18 - 30(evens), 42, 44 and 46# 18 - 30(evens), 42, 44 and 46