Posts Tagged ‘proofs’
Tuesday, December 29th, 2009
How to Use Axioms and Postulates in Proofs
Description
A detailed tutorial on axioms and postulates. Step by step tutorial including several examples of axioms and postulates for reference.
Overview
Axioms, sometimes called postulates, are parts of a proof. When you write a proof, there are several important parts: the given information and statements explaining the given information, the proof itself along with examples, and the conclusion. Axioms and postulates are the first set of statements that contain the given information. These statements are all assumed to be true. An important part of axioms and postulates are undefined terms, from which new concepts can be assumed, and new theorems can be deduced.
Tags: axioms, conclusion, deduced, discrete math, examples, given, information, postulates, proofs, statements, terms, theorems, undefined
Posted in Discrete Math | No Comments »
Tuesday, October 20th, 2009
How to Write Proofs by Exhaustion
Description
A detailed tutorial on writing proofs by exhaustion. Step by step tutorial including several examples of how to write proofs by exhaustion for reference.
Overview
A proof by exhaustion is one of the easier types of proofs to write. All this proof involves is testing cases – every case possible for what you are trying to prove. This can be made easier by using variables instead of numbers, or by testing for an even number and odd number, positive and negative number, etc. That way you do not have to test many numbers in order to prove. If even one of the cases does not work out, then whatever you are testing for has been disproven.
Tags: cases, discrete math, disproven, even, exhaustion, Math, method, negative, odd, positive, possibilities, proof, proofs, proven, variable, write
Posted in Discrete Math | No Comments »
Thursday, October 1st, 2009
Relationship of Deductive Reasoning to Mathematics
Description
A detailed tutorial on the relationship of deductive reasoning to mathematics. Step by step tutorial including several examples of the relationship of deductive reasoning to mathematics for reference.
Overview
Deductive reasoning
, also known as deduction, is a type of reasoning that constructs or evaluates deductive arguments. These arguments are known as proofs, which are very important in math and lead to all types of inductive reasoning. These arguments are said to be either valid or invalid, never true or false.
Tags: arguments, conclusion, deduction, deductive logic, deductive reasoning, facts, invalid, Math, proofs, propositions, theorems, valid
Posted in Math | No Comments »
Thursday, September 24th, 2009
The Modus Tollens Rule Explained
Description
A detailed tutorial on the modus tollens rule. Step by step tutorial including several example problems of the modus tollens rule for reference.
Overview
Modus tollendo tollens, often simply referred to as modus tollens, is an argument in logic that states if P, then Q. Negation of Q, therefore negation of P. This is sometimes called denying the consequent, and is often confused with the indirect proof of proving by contraposition. There are several forms that the modus tollens rule can take, depending on when and how you are using it.
Logical Operator Notation: 
Basic Form: 
With Assumptions: 
Set Theory:
Predicate Logic:
Tags: assumptions, discrete math, logic, logical operator, Math, modus tollendo tollens, modus tollens, negation, P, predicate, proofs, Q, rule, sequent, set theory, then, therefore, truth tables
Posted in Discrete Math | No Comments »
Thursday, September 24th, 2009
The Modus Ponens Rule Explained
Description
A detailed tutorial on the modus ponens rule. Step by step tutorial including several examples of the modus ponens rule for reference.
Overview
Modus ponendo ponens, typically shortened to just modus ponens, is an argument in logic. It is closely related to the argument modus tollens. Modus ponens states that if P, then Q. P, therefore Q. This can be expressed in either sequent form or rule form for formal notation.
Sequent Form: 
Rule Form: 
Tags: discrete math, logic, Math, modus ponendo ponens, modus ponens, P, proofs, Q, rule, sequent, then, therefore, truth tables
Posted in Discrete Math | No Comments »
Thursday, September 24th, 2009
How to Solve Proofs by Contraposition
Description
A detailed tutorial on how to solve proofs by contraposition. Step by step tutorial including several example problems of solving proofs by contraposition for reference.
Overview
The method of writing proofs is not entirely a set process – every mathematician brings their own style to their proof, just like an author will bring their own style to their books. However, there are several different basic techniques for writing proofs. One of these is writing proofs by contraposition. A proof by contraposition is by using negation with the antecedent and consequent. You will state that the consequent is false if declared true, and true if declared false. You will then prove that the antecedent is true if it was declared false, or false if it was declared true. If you can prove the contraposition of the statement, then you can also consider that to be the proof of the statement.
Tags: antecedent, consequent, contraposition, discrete math, false, Geometry, Math, negation, proof, proofs, true
Posted in Discrete Math | No Comments »
Thursday, September 24th, 2009
How to Solve Proofs by Contradiction
Description
A detailed tutorial on how to solve proofs by contradiction. Step by step tutorial including several example problems of solving proofs by contradiction for reference.
Overview
The method of writing proofs is not entirely a set process – every mathematician brings their own style to their proof, just like an author will bring their own style to their books. However, there are several different basic techniques for writing proofs. One of these is writing proofs by contradiction. A proof by contradiction is when you take the antecedent and the consequent, and assume the negation of the antecedent – that is to say, say it is false if it is declared true, and true if it is declared false. Then attempt to prove the consequent. If you cannot prove it, then the statement has been proven.
Tags: antecedent, consequent, contradicition, discrete math, false, Geometry, Math, negation, proof, proofs, true
Posted in Discrete Math | No Comments »
Thursday, September 17th, 2009
An Introduction to Writing Proofs
Description
A detailed tutorial on how to write a simple proof. Step by step tutorial including an example problem on writing simple proofs for reference. Knowledge of proof writing is required in nearly all branches of math and science.
Overview
The art of writing proofs is different than anything else that is done in math, yet it is what math is all about. A proof is like math literature, it is taking something that we already know to be true or false and proving why it is true or false. Typically variables are used with many algebra tricks, although in geometry proofs you are required to write paragraphs on certain things. Proofs all have a different style depending on who wrote them – it is the one part of math you are allowed to use your creativity in. Remember, there is no “right” and “wrong” with proofs, provided you either prove or disprove what the problem asked you to.
Tags: discrete math, disprove, geometrical proofs, literature, Math, proofs, prove, proven, two column proofs, writing, writing proofs
Posted in Discrete Math | No Comments »
