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Inductive Reasoning and Logic

Conditional Statements

Angle and Segment Addition

Deductive Reasoning

Postulates and Proof

Inductive Reasoning and Logic for $100

Define: Inductive Reasoning

Answer

Inductive Reasoning: Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction

Back

Inductive Reasoning and Logic for $200

Define: Conjecture

Answer

Conjecture: An educated guess based on known information

Back

Inductive Reasoning and Logic for $300

Fill in the blanks:The _______ of a statement

is whether a statement is true or false. A statement is false if one __________ can be found.

Answer

The Truth Value of a statement is whether a statement is true or false. A statement is false if one Counterexample can be found.

Back

Inductive Reasoning and Logic for $400

Write a logical statement using a disjunction and a negation

Answer

~PvQ

Back

Inductive Reasoning and Logic for $500

Draw a truth table for the following compound statement:

(Pv~Q)^R

Answer

Back

P Q ~Q (Pv~Q) R (Pv~Q)^R

T T F T T T

T T F T F F

T F T T T T

T F T T F F

F T F F T F

F T F F F F

F F T T T T

F F T T F F

Conditional Statements for $100

Identify the hypothesis and conclusion in the following conditional statement:

If we are playing Jeapordy then, the students try to talk too much after questions

Answer

Hypothesis: We are playing Jeapordy

Conclusion: The students try to talk to much after questions

Back

Conditional Statements for $200

Write the following statement as a conditional in if-then form:

Studying Descartes makes students happy

Answer

If students are studying Descartes, then they are happy

Back

Conditional Statements for $300

Determine whether or not the following statement is reversible (i.e. whether or not the converse is true). If it is, write the statement as a biconditional:

Two lines that are skew are not coplanar.

Answer

No, they could be parallel.

Example of biconditional:

A triangle is a polygon with 3 sides

It is a triangle if and only if it is a polygon with three sides

Back

Conditional Statements for $400

What is logically equivalent to the converse of a statement?

Answer

Back

The Inverse of the same statement

Conditional Statements for $500

Write the conditional, converse, inverse, and contrapositive of the following statement and determine the truth value for each:

All squares are rectangles

Answer

Back

Conditional: If it is a square then it is a rectangle. True

Converse: If it is a rectangle then it is a square. False

Inverse: If it is not a square then it is not a rectangle. False

Contrapositive: If it is not a rectangle then it is not a square. True

Deductive Reasoning for $100

Define: Deductive Reasoning

Answer

Deductive Reasoning- Uses facts, rules, definitions, or properties to reach logical conclusions

Back

Deductive Reasoning for $200

Write the law of Detachment:

Answer

If p implies q is true and p is true, then q is also true

[(p→q)^p]→q

Back

Deductive Reasoning for $300

Write the law of Syllogism

Answer

If p implies q and q implies r are true, then p implies r is also true.

[(p→q)^(q→r)]→(p→r)

Back

Deductive Reasoning for $400

Use the law of detachment to draw a conclusion from the two statements given:

1)I can go to the concert if I can afford to buy a ticket

2)I can go to the concert

Answer

No conclusion possible:

Conditional: If I can afford to buy a ticket, then I can go to the concert

Thus, (p→q)^q => Not the Law of Detachment: [(p→q)^p]→q

Back

Deductive Reasoning for $500

Use the Law of Syllogism to draw a conclusion from the two statements given:

1) If a number is a multiple of 64, then it is a multiple of 8

2) If a number is a multiple of 8, then it is a multiple of 2

Answer

If a number is a multiple of 64, then it is a multiple of 2 by the Law of Syllogism

[(p→q)^(q→r)]→(p→r)

Back

Postulates and Proof for $100

Define:

Paragraph (Informal) Proof

Answer

Back

Paragraph (informal) Proof – A paragraph that explains why a conjecture for a given situation is true.

Postulates and Proof for $200

Write a Paragraph Proof showing that given that M is the midpoint of PQ, then PM MQ

Answer

From the definition of midpoint of a segment, PM = MQ. This means that PM and MQ have the same measure. By the definition of congruence, if two segments have the same measure, then they are congruent. Thus PM MQ

Back

Postulates and Proof for $300

Write a two column proof showing that:

If AB = BC;AB =-5x+3;BC = -4+2x

Then, x = 1

Answer

Back

Statements ReasonsAB = BC; AB = -5x + 3

BC = -4 + 2xGiven

-5x + 3 = -4 + 2x Substitution+5x – 5x + 3 = -4 + 2x + 5x Addition Prop. of Equality3 = -4 + 7x Simplify3 + 4 = -4 + 4 + 7x Addition Prop. of Equality7 = 7x Simplify7/7 = (7x)/7 Division Prop. of Equality1 = x Simplifyx = 1 Symmetric Prop. of Algebra

Postulates and Proof for $400

Given that (5x-2)/2 = 9, write a two column proof to show that x=4

Answer

Back

Statements Reasons

(5x-2)/2 = 9 Given

2* (5x-2)/2 = 9 *2 Multiplication Prop of Equality

5x-2 = 18 Simplify

5x-2+2 = 18 + 2 Addition Prop. of Equality

5x = 20 Simplify

(5x)/5 = 20/5 Division Prop of Equality

x = 4 Simplify

Postulates and Proof for $500

List the 7 Axioms of Geometry

Answer1) Through any two points, there is exactly one line2) Through any three non-collinear points there is

exactly one plane3) A line contains at least two points4) A plane contains at least 3 points5) If two points lie in a plane, then the entire line

containing those points lies in that plane6) If two lines intersect, then their intersection is

exactly one point7) If two planes intersect then their intersection is

line

Back

Angle and Segment Addition for $100

Name the Property of congruence of angles that justifies the statement:

If Angle 1 Angle 2,

then Angle 2 Angle 1

Answer

Symmetric Property

Back

Angle and Segment Addition for $200

Name the Postulate that justifies the statement:

Given: Points A,B,C are collinear and point B lies between A and C, then

AB + BC = AC

Answer

Segment Addition Postulate

Back

Angle and Segment Addition for $300

Given that <PQR and <RQS form a linear pair, solve for x

R

7x-323x+22

PQ S

Answer

7x – 32 + 3x + 22 = 180

10x – 10 = 180

10x = 190

x = 19

Back

Angle and Segment Addition for $400

Write a two column proof showing that if AC = 40 and points A, B,C are collinear, then x = 4

AnswerStatements Reasons

AC = 40;A,B,C are Collinear

AB = 2x; BC = 6x+8

Given

AB + BC = AC Segment Addition Postulate

2x + 6x + 8 = 40 Substitution

8x + 8 = 40 Combining Like Terms/Simplify

8x + 8 – 8 = 40-8 Subtraction Prop. of Equality

8x = 32 Simplify

(8x)/8 = 32/8 Division Prop. of Equality

X = 4 Simplify Back

Angle and Segment Addition for $500

Given that m<PQR =7+x; m<RQS = 52 and m<PQS = 2x + 8. Find x

Answer

Back

Using the Angle addition postulate:m<PQR + m<RQS = m<PQS

(7+x) + 52 = 2x + 8

x + 59 = 2x + 8

x + 51 = 2x

51 = x