How to Write Proofs

Features

DC Proof is a free, downloadable, PC-based educational software program that can be used to learn basic proof-writing skills

Recommended for use by...

undergraduates in transition to proof-based mathematics

teachers in training

advanced high school students

Simplified rules and axioms of logic and set theory

based on common practices and simplifying assumptions implicit in most mathematics textbooks at the undergraduate level

Color-coded variables indicate at a glance what generalizations can be made on them

Convenient, pull-down menus enables you to apply any of the rules and axioms of logic, set theory and number theory with the click of a mouse

   Fig. 1. Menu of the rules of Logic

See additional screen shots

User comments and meaningful variable names (of unlimited length) facilitate both the reading and writing of proofs

Suppose that it is sunny if any only if it is it not cloudy.

1     Sunny <=> ~Cloudy
     Premise

Fig. 2 User comment in blue. Line number and logical statement in black. Rule used in gray.

Multi-level views of proofs that hide or reveal details as required ─ indispensible for writing longer proofs!

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Symbolic Logic and The Basic Methods of Proof

DC Proof includes an interactive tutorial that introduces symbolic logic and the basic methods of proof using worked examples and exercises with hints and full solutions

Topics in symbolic logic include:

Logical propositions and predicates

Logical operators

Quantifiers ─ free and bound variables

Methods of proof include:

Direct proof

Indirect proof

Proof by contradiction

Proof by contrapositive

Proof by cases

Proof of biconditionals

Proof by induction

Use of the formal axioms and definitions

Set theory

Elementary number theory

Examples of the Methods of Proof

Tutorial Example Direct
Proof Proof by
Contradiction Proof by
Contrapositive Proof by
Cases Proof of Biconditionals Manipulating
Quantifiers Proof by Induction Set Theory Number Theory

1. The Commutativity of AND
x

2. The Transitivity of IMPLIES
x

3. The Commutativity of OR (by contradiction) x

4. The Commutativity of OR (by contrapositive)

x

5. Proving a Biconditional

x
x x

6. The Distributivity of AND over OR (Part 1)

x
x

7. The Transitivity of Equality x x

8. The Composition of Functions x x

9. The Barber Paradox x x x

10. The Paradox of the Universal Set x x x

11. The Uniqueness of the Empty Set x x x x x x

12. 1+1=2, 2+2=4 x x x x

13. No Number its own Successor x x x x x

Fig. 3 Table of methods of proof used in tutorial

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Tutorial Example	Direct Proof	Proof by Contradiction	Proof by Contrapositive	Proof by Cases	Proof of Biconditionals	Manipulating Quantifiers	Proof by Induction	Set Theory	Number Theory
1. The Commutativity of AND	x
2. The Transitivity of IMPLIES	x
3. The Commutativity of OR (by contradiction)		x
4. The Commutativity of OR (by contrapositive)			x
5. Proving a Biconditional	x	x			x
6. The Distributivity of AND over OR (Part 1)	x			x
7. The Transitivity of Equality	x					x
8. The Composition of Functions	x					x
9. The Barber Paradox		x				x		x
10. The Paradox of the Universal Set		x				x		x
11. The Uniqueness of the Empty Set	x	x	x		x	x		x
12. 1+1=2, 2+2=4	x					x		x	x
13. No Number its own Successor		x				x	x	x	x