Download Chapter 2 Notes Niven – RHS Fall 12-13

Chapter 2 Notes
Niven – RHS
Fall 12-13
Inductive reasoning is when you find a pattern is specific cases and then write a conjecture for the general case.
A conjecture is an unproven statement that is based on observations.
Inductive reasoning boils down to analyzing a given set of data or observations, recognizing patterns, and making a
generalization, called a conjecture.
If you want to show that a conjecture is false (or disprove the conjecture) you only need to show ONE example
where the conjecture does not work. That example proving a conjecture false is called a counterexample.
A counter example is a specific case for which the conjecture is false.
To show that a conjecture is true you must show that it is true for all cases. You do this using deductive reasoning.
A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion.
When a conditional statement is written in the “if-then” form, the if clause is the hypothesis, and the then clause is
the conclusion.
The negation of a statement is the opposite of the original statement. If a statement is already negative, then the
negation will be positive.
Every conditional statement has three related conditionals. These are the converse, the inverse, and the
contrapositive.
The converse is formed by interchanging the hypothesis and the conclusion.
The Inverse is created by negating the hypothesis and conclusion.
The contrapositive is formed by interchanging AND negating the hypothesis and conclusion.
Original Conditional:
If hypothesis, then conclusion.
Converse:
If conclusion, then hypothesis.
Inverse:
If NOT hypothesis, then NOT conclusion.
Contrapositive
If NOT conclusion, then NOT hypothesis.
When two statements have the same truth value (meaning that they are both true or both false) then they are said
to be logically equivalent. They are said to be equivalent statements.
You can write definitions as a conditional statement in the if then form or as its converse. If it is a definition then the
original conditional statement and it converse will be true.
When a conditional statement and its converse are both true, you can write them as a single bi-conditional
statement. A bi-conditional statement contains the phrase if and only if.
Deductive reasoning uses facts, definitions, accepted properties, and the laws of logic to form a logical argument.
The law of detachment is also called direct
argument.
Law of Syllogism is very similar to the
algebraic property called the Transitivity
Property.
EXAMPLES:
If two angles have the same measure, then they are congruent. mA = mB.
Josh goes to the gym every weekday. Today is Wednesday.
If Rick takes chemistry this year, then Jesse will be Rick’s lab partner.
If Jesse is Rick’s lab partner, then Rick will get an A in chemistry.
If 𝑥 2 > 25, then 𝑥 2 > 20.
If 𝑥 > 5, then 𝑥 2 > 25.
We will use these laws of logic when writing proofs.
In geometry, rules that are accepted without proof are called postulates or axioms. Rules that are proved are called theorems.
An argument that follows deductive reasoning is called a proof. A conjecture that has been proven is called a theorem. A proof
transforms a conjecture into a theorem.
Postulates and theorems are often written in conditional form. Remember that postulates and theorems are different from
definitions and we cannot also write postulates and theorems as bi-conditional statements.
When you solve an equation, you use properties of real numbers. Segment lengths and angle measures are real numbers, so
you can also use these properties to write logical arguments about geometric figures. The following properties of equality are
true for segment lengths and angle measures.