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Chapter 1: Problem 15

What is the difference between a postulate and a definition?

Short Answer

Expert verified
A postulate is assumed true without proof, while a definition explains a term's meaning.

Step by step solution

01

- Understanding a Postulate

A postulate is a statement assumed to be true without proof. It serves as a starting point for further reasoning and arguments in mathematics or logic. For example, in geometry, one common postulate is that through any two points, there is exactly one straight line.
02

- Understanding a Definition

A definition is a statement that explains the meaning of a term or concept. It is used to clarify what is being discussed and to set the boundaries of that concept. For example, the definition of a triangle is a polygon with three sides.
03

- Comparing Postulates and Definitions

To compare the two, note that a postulate is assumed without proof and often serves as a foundational element for building logical arguments. In contrast, a definition explains the meaning of a term and is used to describe specific properties or characteristics.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Postulate
A postulate is a statement that we accept as true without requiring proof. It forms the backbone of further reasoning and arguments in subjects like mathematics and geometry. For instance, one key geometric postulate is that a single straight line can pass through any two points. Postulates act as starting points in logical sequences and help build complex concepts.

Think of postulates as the ground rules of a game. They must be accepted by everyone playing, even if we don't explain why they are true.
Definition
A definition, on the other hand, is a statement that clearly explains what a term or concept means. It sets the boundaries and specifics of the term so everyone understands exactly what is being discussed. For example, the definition of a triangle tells us it has three sides. Definitions are essential for communication, ensuring that when we talk about a concept, everyone knows precisely what it entails.

Definitions can be thought of as detailed markers on a map. They tell you the specific features of each location, so there's no confusion.
Geometry Concepts
Geometry is full of postulates and definitions that help us understand shapes, sizes, and dimensions. Some crucial geometry concepts include basic shapes like triangles, squares, and circles, as well as concepts like points, lines, and angles.

Here's how these concepts fit together:
  • Points: The most basic unit with no size, only a position.
  • Lines: Extend infinitely in both directions and contain an infinite number of points.
  • Angles: Formed by two rays (parts of lines) coming out from the same point.

    Understanding these concepts starts with knowing the foundational postulates and definitions that describe them. This foundation helps us build a complete picture of geometric principles.

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Most popular questions from this chapter

The puzzles are classic examples and a certain amount of deductive reasoning is required to solve them. Some of these puzzles are quite challenging, so don't be discouraged if you have trouble finding the solution immediately. Ideally they will make you think a bit and, along the way, provide a bit of entertainment. A museum fired an archaeologist who claimed she found a coin dated 300 B.C. Why?

What is the supplement of the supplement of an angle measuring \(160^{\circ} ?\)

For problems 13-26, explain the reasoning in one or two complete sentences. If \(\angle A\) and \(\angle B\) are complementary, can \(\angle A\) and \(\angle B\) be vertical angles too?

The puzzles in Exercises have all been attributed to Englishman Henry Ernest Dudeney \((1857-1930),\) called by some the greatest puzzle writer of all time. Consider the following six "postulates." a. Art, Bev, Cheryl, Dot, and Ed all attend the same college where one is a freshman, one is a sophomore, one is a junior, one is a senior, and one is a graduate student. b. Art, Bev, and Cheryl have not yet completed their undergraduate work. c. Bev is one year ahead of Ed. d. Art is not a freshman. e. Ed is not a freshman. f. Art is in a higher class than Ed. Can you discover theorems that give each student's class in college? Write the theorems in complete sentences.

The puzzles in Exercises have all been attributed to Englishman Henry Ernest Dudeney \((1857-1930),\) called by some the greatest puzzle writer of all time. Consider the following eight "postulates." a. Smith, Jones, and Rodriquez are the engineer, brakeman, and fireman on a train, not necessarily in that order. b. Riding on the train are three passengers with the same last names as the crewmembers, identified as passenger Smith, passenger Jones, and passenger Rodriquez in the following statements. c. The brakeman lives in Denver. d. Passenger Rodriquez lives in San Francisco. e. Passenger Jones long ago forgot all the algebra that he learned in high school. f. The passenger with the same name as the brakeman lives in New York. g. The brakeman and one of the passengers, a professor of mathematical physics, attend the same health club. h. Smith beat the fireman in a game of tennis at a court near their homes. Can you discover a theorem that tells the name of the engineer? The brakeman? The fireman? Write the theorems in complete sentences.
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