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Definition
proof
/pruːf/noun
Meaning
A sequence of logical steps that forces a conclusion to be true given the starting assumptions, with no room left for doubt or alternative interpretation.
See the full entry →
Concept
Concept · Logic
Mathematical proof
in the context of Reasoning
A chain of undeniable steps that forces someone to accept a conclusion.
unpack it →
Lesson
Lesson

What Makes A Statement Actually True

subject: Learn how implications survive the morning after calculus.
TEST THE CONDITIONAL STATEMENTS
Key Point
The Key Point
Every proof works by connecting something you know to something you want to show is true, using only rules that force the connection to hold.
Walk through how this works →
Lesson
Lesson

Assume The Opposite And Break Math

subject: See how indirect arguments sneak up on certainty.
WALK THROUGH A CONTRADICTION
Bullets
Outline
Ways people usually start a proof
  • ▸Assume the opposite is true
  • ▸Build it up from scratch
  • ▸Work backward from the goal
see the outline →
Lesson
Lesson

Why Converse Statements Fail Every Time

subject: Stop turning arrows backward when nobody is looking.
SPOT THE REVERSED IMPLICATION
Key Point
The Key Point
A calculation gives you an answer; a proof gives you permission to stop doubting it.
See what changes between them →
Lesson
Lesson

Counting Infinities Without Losing Your Keys

subject: Watch hotel rooms fill up while the lobby stays empty.
BUILD THE INFINITE LIST
Lesson
Lesson

Building Steps That Scale Up Forever

subject: Climb the domino ladder without dropping the base case.
TRACE THE INDUCTION DOMINOES
Related Concepts
made oftype ofmade ofruns onAssumptionsExamplesSymbolsLogicProof
explore map
Lesson
Lesson

Translating English Into Cold Logic Symbols

subject: Strip the adjectives until only the quantifiers remain.
TRANSLATE THE EVERYDAY SENTENCE
Analogy
Analogy · Navigation
Unfamiliar
A proof
Is Like
→
Familiar
Directions you can verify at each step
Explore how the analogy holds →3 pairs
Lesson
Lesson

When Proofs Accidentally Prove Too Much

subject: Catch the tiny slip that lets all horses become the same color.
FIND THE BROKEN STEP
Lesson
Lesson

Proving Something Exists Without Showing It

subject: Find the needle in the haystack by counting empty boxes.
USE THE EXISTENCE ARGUMENT
Book

When Math Broke Its Own Rules

The 1931 Logical Shock That Proved Certain Truths Can Never Be Proven

Open the book
Key takeaways
  • The 1931 Logical Shock That Proved Certain Truths Can Never Be Proven
Lesson
Lesson

Breaking Down Cases One Door At A Time

subject: Examine every exit when the hallway splits in three.
CHECK ALL POSSIBLE CASES
Worked Example
Worked Example · Algebraic proof by definition
Problem

Show that if n is an even number, then n² is also even. You can use the fact that even means divisible by 2, and the basic rules of multiplication.

Work through the steps →4
Lesson
Lesson

Reading Greek Letters Like Old Friends

subject: Pronounce your epsilons before handing them to an auditor.
READ THE EPSILON STATEMENT
Practice
Practice

Sorting Good Logic From Clever Nonsense

Test your patience on sentences that almost hold water.

SORT THE VALID CLAIMS
Concept
Concept · Number Theory
Infinite primes
in the context of Number Theory
Euclid's neat little trap that shows you can never run out of prime numbers.
unpack it →
Practice
Practice

Matching Premises To Their Proper Conclusions

Connect the loose ends before the argument unravels.

PAIR PREMISES WITH RESULTS
Practice
Practice

Spotting Hidden Assumptions In Plain Sight

Catch the unwritten rules hiding in the second paragraph.

FIND THE MISSING PREMISE
Worked Example
Worked Example · Reductio ad absurdum
Problem

Show that √2 is not a rational number (cannot be written as a fraction of two integers). Assume the opposite and see what breaks.

See how the contradiction emerges →5
Practice
Practice

Flipping Conditionals Without Getting Dizzy

Practice contrapositives until they feel like second nature.

INVERT THE CONDITIONAL STATEMENTS
Practice
Practice

Stepping Through Simple Induction Drills

Push the first domino and watch the rest fall.

RUN THE INDUCTION STEPS
Bullets
Outline
Simple exercises to build proof muscles
  • ▸Prove an even plus even is even
  • ▸Test if a sum of squares works
  • ▸Spot the missing step in a draft
see the outline →
Practice
Practice

Hunting Flaws In Famous False Proofs

Discover where the algebra broke under the rug.

CATCH THE HIDDEN DIVISION
Warning
Warning
You assume the conclusion partway through to make the steps easier, then act surprised when you arrived where you started. The logic locks shut and no reader can escape it, because you built the loop yourself.
See how to spot it in your own work →
How To
How To · FIG. 10

Writing Your Very First Direct Proof

Take two even numbers and walk them down the aisle.

  1. Define the terms
  2. Substitute the expressions
  3. Conclude the result
DRAFT THE DIRECT PROOF
How To
How To · FIG. 10

Drafting A Proof By Contradiction

Assume the worst and watch the equations crash.

  1. Assume the negation
  2. Derive an absurdity
  3. Claim the original
BUILD A CONTRADICTION
Concept
Concept · Notation
Math symbols
in the context of Mathematical Notation
Shorthand abbreviations that save space once you learn how to read them.
unpack it →
How To
How To · FIG. 9

Setting Up Induction Without Panic

Prove the base case before trusting the machine.

  1. Verify base case
  2. Assume kth step
  3. Prove k plus one
SETUP THE INDUCTION LOOP
Quiz Mc
Understanding the boundary · 2
Question
You show that two quantities are equal by plugging in numbers and testing five examples where they match. Is that a proof?
Yes, if you test enough examples it becomes a proof
No, examples can only suggest a pattern, not guarantee it holds everywhere
Yes, but only if the examples are carefully chosen
Only if you test infinitely many examples
Take the quiz →
How To
How To · FIG. 10

Checking Your Proof For Circular Reasoning

Make sure your conclusion didn't sneak into your premises.

  1. Isolate the core claim
  2. Trace dependency chains
  3. Remove circular loops
AUDIT YOUR OWN LOGIC
How To
How To · FIG. 7

Formatting Math Text Like A Professional

Align your equals signs before showing the professor.

  1. Align equals signs
  2. Indent major steps
  3. State final conclusions
FORMAT THE FINAL DRAFT
Quiz Mc
Valid moves in the logical world · 2
Question
In a proof, you write: 'Let x be any positive integer. Then x + 1 is also a positive integer.' Does this work as a starting point?
Yes, you are just stating obvious facts
Yes, this sets up a variable to represent any member of a class
No, you cannot use variables in proofs
Only if you list out actual examples first
Take the quiz →
Reference
QA 12 .O3
Reference

The Glossary Of Mathematical Symbols

Look up what upside-down letters mean in plain English.

see also: see also: notation, syntax, shorthand
BROWSE THE SYMBOL GLOSSARY
Reference
HD 880 .J6
Reference

Standard Proof Techniques At A Glance

Review the shelf of methods before picking your tool.

see also: see also: direct, contrapositive, induction
CONSULT THE METHOD LIST
Reference
QA 171 .E3
Reference

Common Fallacies That Ruin Good Proofs

Check your draft against the graveyard of bad logic.

see also: see also: circularity, converse, ambiguity
VIEW THE FALLACY ROSTER
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