Mathematics Definitions Chapter 1.4

Questions and Answers

What is a definition?

A statement that describes the qualities of an idea, object, or process.

What is a postulate?

A statement that is assumed to be true without proof.

What is a common notion?

A statement that is not officially defined but is understood to be common sense.

What is a two-column proof?

A type of proof that has two columns: one for statements and one for reasons. Signup and view all the answers

What is an indirect proof?

A type of proof written in paragraph form showing that the contradiction of a statement is false. Signup and view all the answers

What is a flowchart proof?

A type of proof that uses a graphical representation with statements in boxes. Signup and view all the answers

Two-column proof and flowchart proof both present a logical flow of deduction statements each paired with its justification.

True Signup and view all the answers

What is a theorem?

A statement that has already been proven to be true. Signup and view all the answers

What is a corollary?

A statement that makes sense based on a statement that has already been proven. Signup and view all the answers

A corollary _____ a theorem.

proves Signup and view all the answers

What is an example of a postulate?

A line segment can be drawn between any two points. Signup and view all the answers

Why does one need to have definitions in proofs?

They allow all people involved in the proof to know they are talking about the same things. Signup and view all the answers

What is the process of proving?

Step 1: Use definitions, postulates, and common notions to prove a statement; Step 2: The statement becomes a theorem; Step 3: Use the theorem to prove corollaries. Signup and view all the answers

Study Notes

Definitions and Key Concepts

Definition: A precise statement describing the qualities of an idea, object, or process, vital for proofs.
Postulate: An assumed statement that is accepted as true without proof; serves as a foundational rule in proofs.
Common Notion: An intuitive statement understood as common sense, which can be utilized in proofs without formal definition.

Types of Proofs

Two-Column Proof: Organized into two columns; left side for statements or deductions and right side for justifications (definitions, postulates, or theorems).
Indirect Proof: Presented in paragraph form, proving the original statement by showing that its negation leads to a contradiction; also known as proof by contradiction.
Flowchart Proof: Utilizes a graphical format where statements are in boxes and justifications are beneath them, indicating logical relationships with arrows.

Relationships Between Theorems and Statements

Theorem: A statement that has been proven true through deductive reasoning, establishing a basis for further proofs.
Corollary: A statement that follows logically from an already proven theorem, reinforcing established truths.

Purpose and Importance of Definitions

Definitions ensure clarity in proofs, allowing all participants to have a consistent understanding of the terms and concepts being used.

Proofing Process Overview

Step 1: Utilize definitions, postulates, and common notions to establish the truth of a statement.
Step 2: Successfully proven statements become theorems, recognized as consistently true.
Step 3: Use theorems to demonstrate corollaries and other related statements.

Example Postulate

A fundamental example of a postulate is: "A line segment can be drawn connecting any two points," illustrating a basic geometric principle.

Description

Test your knowledge of key mathematical definitions including terms such as definition, postulate, and common notion from Chapter 1.4. This quiz helps reinforce your understanding of essential concepts used in proofs and mathematical reasoning.