CMClearMathAcademy

Good Definitions vs. Bad Definitions

A free Logic lesson from the “Biconditionals and Definitions” unit, with a worked example and practice problems including step-by-step solutions.

Definitions should classify examples correctly. A bad definition may include things it should exclude, exclude things it should include, or use the term it is trying to define. Learning objective: Identify definitions that are too broad, too narrow, or circular. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: 'x is even iff x is divisible by 2' works in both directions. Example 2: A definition that only works one way is too broad or too narrow. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: Definitions in math and science depend on knowing when a condition works both ways.

Worked example

Problem. Example case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. A biconditional is two-way.
  4. It contains both conditional directions.

Answer: p implies q and q implies p

Practice problems

1. Practice case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?

Choices: p implies q and q implies p · p implies q only · p and q are both false · not p or q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. A biconditional is two-way.
  4. It contains both conditional directions.
  5. So p ↔ q means each statement implies the other.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: p implies q and q implies p

2. Practice case B (Good Definitions vs. Bad Definitions): Which definition works as a biconditional?

Choices: A number is even iff it is divisible by 2. · A number is even iff it is positive. · A rectangle is a square iff it has four sides. · A prime number is any odd number.

Show solution
  1. A good definition works both ways.
  2. Even numbers are exactly the integers divisible by 2.
  3. The other definitions are too broad or false.

Answer: A number is even iff it is divisible by 2.

3. Practice case C (Good Definitions vs. Bad Definitions): A definition is too broad when it:

Choices: includes things that should be excluded · excludes correct examples · uses symbols · has examples

Show solution
  1. Too broad means the category catches extra objects.
  2. For example, defining square as any four-sided figure includes rectangles that are not squares.
  3. So it includes too much.

Answer: includes things that should be excluded

4. Practice case D (Good Definitions vs. Bad Definitions): A definition is too narrow when it:

Choices: excludes things that should be included · includes every possible object · uses if and only if · has a non-example

Show solution
  1. Too narrow means real examples are left out.
  2. A good definition includes all and only the intended objects.
  3. So the first choice is correct.

Answer: excludes things that should be included

5. Practice case E (Good Definitions vs. Bad Definitions): Which object is a non-example for "a square is a rectangle with four equal sides"?

Choices: A 3 by 5 rectangle · A 4 by 4 square · A square tile · A rectangle with all sides equal

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Good Definitions vs. Bad Definitions): Which object is a non-example for "a square is a rectangle with four equal sides"?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. A non-example should fail the definition.
  4. A 3 by 5 rectangle does not have four equal sides.
  5. So it is not a square.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: A 3 by 5 rectangle

Practice this interactively with instant feedback and an AI tutor.

Practice Good Definitions vs. Bad Definitions Take the free placement check

More Logic lessons