Chapter 2: Logical Arguments and Line Relationships

2.1: Conjectures and Counterexamples

Vocabulary

A conterexample statement where the hypothesis is the same but conclusion is different.

Inductive reasoning is reasoning with examples

A conjecture is a concluding statement based on inductive reasoning

Example of Conterexample

Hypothesis: The lost planet is round.

Conclusion: The planet is Neptune as Neptune is round.

Conterexample: Mars is also round.

Since a conterexample was provided, the conclusion is false.

Example of Conjectures

Think about this pattern: 0, 5, 10, 15, 20

A conclusion or conjecture can be made that it incresses by 5.

2.2: Statements, Conditionals, and Biconditionals

Vocabulary

A truth value is whether or not a statement is true based on the given information.

A truth table is a table where truth values are organized.

Logical Operators

Name	Symbol
^	And
v	Or
~	Not

Truth Tables

a: Mars is red

b: Neptune is green

a	b	a v b	a ^ b	~b
true	false	true	false	false

2.3: conditional Statements

Vocabulary

A conditional statement is written in a if then statement.

Switching the if and then makes it a converse of the statement.

An inverse of a statement is when you take the oposite of the if and the opposite of the then.

A contrapositive is the inverse of the converse.

Python Examples of Vocabulary

# fact
a = True
b = False

# conditional statement
if a:
  b = False

# converse
if not b:
  a = True

# inverse
if not a:
  b = not False # aka True

# contrapositive
if b:
  a = not True # aka False

All of these ifs produce the same result!!!

Identifying the Hypothesis and Conclusion

The hypothesis is the statement that follows the if while the conclusion follows the then.

For example:

if a:
  b = True

In this example a or a == True is the hypothesis and b = True is the conclusion.

2.4: Deductive Reasoning

Vocabulary

Deductive reasoning is when a logical conclusion is reached from using facts, rules, definitions, or properties.

The Law of Detachment is a valid form of deductive reasoning with an if then format and a conclusion.

The Law of Syllogism is a valid form of deductive reasoning which can be used to pice together two or more statements in which one has a conclusion of the other hypothesis.

Examples

Given: If you ride to school then you have a bike. Bob rides to school.

Conclusion: Bob has a bike

This is a valid conclusion made using the Law of Detachment.

Note

If the converse of the given hypothesis was given, then the conclusion would be invalid. This is because the fact code Bob rides to school would address the conclusion instead of the hypothesis.

2.5: Postulate and Paragraph Proof

Vocabulary

An axiom is a statement that is accepted without proof.

An postual is the same as an axiom.

A paragraph proof is a paragraph to explain why a conjecture is true.

Proofs

1. Though any two points, there is one line.

2. Though any three points, there is one plane.

3. A line contains at least two points

4. A plane contains as least three points

5. If two points of a line are in a plane, the line is on the line.

2.6: Algebraic Proof

Vocabulary

The property of equality states that if the same operation is conducted on both sides of an equation on the same number, then both sides of an equations are equal.

The addition property of equality states that if a number is added to both sides of an equation, then both sides of an equation are equal.

The subtraction property of equality states that if a number is subtraction from both sides of an equation then both sides of an equation are equal.

The multiplication property of equality states that if a number is multiplied to both sides of an equation, then both sides of the equation remain equal.

The division property of equality states that if both sides of an equation are divided by the same number, then both sides of the equation remain equal.

The symmetric property of equality states that an equation can be flipped

The transitive property of equality states that sates that if two equations contain a common variable, they can be joined by the common variable.

The substitution property of equality states that a value can be replaced by its alterative value

The distributive property states that a value being multiplied on a sum of difference can be multiplied to each individual number.

The two-column proof is a proof organized into two columns, statements and reasons.

Example of Two-Column Proof

Given: \(frac{5(x - 2) + 7}{12} = 7\)

Prove: \(x = 17.4\)

To write this in two column format, you need to specify every step for finding x aa.

Statement	Reason
\(\frac{5(x - 2) + 7}{12} = 7\)	Given
\(5(x - 2) + 7 = 84\)	Multiplication Property of Equality
\(5(x - 2) = 77\)	Subtraction Property of Equality
\(5x - 10 = 77\)	Distributive Property
\(5x = 87\)	Addition Property of Equality
\(x = 17.4\)	Division Property of Equality

2.7: Segment Relationships

Vocabulary

The reflexive property of congruence states that a line segment is congruent to itself.

The symetric property of congruence states that if a segment is congruent to another segment, the other segment is congruent to that segment.

The transitive property of congruence states that a line segment that is congruent to another can be replaced by that segment.

The ruler postulate states that points can be given numbers.

The segment adddition postulate states that if three points are colinner, The two smaller distances can be added up to form the bigger distence.

2.8: Slope and Equations of Lines

Vocabulary

The definition of congruent angles states that perpendicular lines form a \(90^{\circ}\) angle.

The definition of supplementary angles states that supplementary angles form a \(180^{\circ}\)

The definition of complementary angles states that complementary angles form a \(90^{\circ}\)

2.9: Proving Lines Parallel

2.10: Perpendiculars and Distance