Posts Tagged ‘proof’
Thursday, October 29th, 2009
Overview of Symmetric Relations
Description
A detailed tutorial on the property of symmetric relations. Step by step tutorial including several examples of symmetric relations for reference.
Overview
A symmetric relation can be mathematically defined as for all x, y, and z belonging to A, if x R y and y R z, then x R z. In this statement, A is a set, and R is a relation of that set. An empty set is considered to be symmetric. Since a symmetric relation is defined by a conditional sentence, a proof for the symmetric property of relations would be written as a direct proof.
Tags: conditional, direct, discrete math, empty, equal, equivalence, married, odd, proof, property, r, relation, set, symmetric, x, y
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Transitive Relations
Description
A detailed tutorial on the property of transitive relations. Step by step tutorial including several examples of transitive relations for reference.
Overview
A transitive relation can be mathematically defined as for all x and y belonging to A, if x R y, then y R x. In this statement, A is a set, and R is a relation of that set. An empty set is considered to be transitive. Since a transitive relation is defined by a conditional sentence, a proof for the transitive property of relations would be written as a direct proof.
Tags: conditional, direct, discrete math, divides, empty, equal, equivalence, great, greater, implies, proof, property, r, relation, set, subset, transitive, x, y, z
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Reflexive Relations
Description
A detailed tutorial on the property of reflexive relations. Step by step tutorial including several examples of reflexive relations for reference.
Overview
A reflexive relation can be mathematically defined as for all x belonging to A, x R x. In this statement, A is a set, and R is a relation of that set. If the relation is an empty set, then it is not reflexive, unless the set itself happens to be an empty set. When writing a proof for a reflexive relation, you must attempt to prove that (x, x) does not belong to R. If you cannot prove this, then you know that the relation must be reflexive.
Tags: discrete math, divide, empty, equal, equvalence, greater, less, proof, property, r, reflexive, relation, set, subset, x
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 8th, 2009
Introduction to the Principle of Mathematical Induction
Description
A detailed tutorial of the principle of mathematical induction. Step by step tutorial including several examples of the principle of mathematical induction for reference.
Overview
The principle of mathematical induction is basically a method of proof-writing, which involves trying to prove that a certain statement is true for all natural numbers. The first statement will be proved, and then the next statement, and the next one. In this way, it is similar to a proof by exhaustion. However, since the statement must be proven for all numbers, eventually an integer will be used in the calculations. This should not be confused with mathematical induction – the principle of mathematical induction is actually a type of deductive reasoning.
Tags: deductive, discrete math, exhaustion, induction, interger, k, Math, mathematical, n!, natural, number, principle, proof, reasoning, statement
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 »
