Posts Tagged ‘negation’
Thursday, October 29th, 2009
Overview of Quasitransitive Relations
Description
A detailed tutorial on the property of quasitransitive relations. Step by step tutorial including several examples of quasitransitive relations for reference.
Overview
A quasitransitive relation can be mathematically defined as for all x, y, and z belonging to A, if x R y, y R z, ~(y R x), and ~(z R y), then x R z and ~(z R x). In this statement, A is a set, and R is a relation of that set. A quasitransitive relation is considered to be a weak version of a transitive relation. If the relation also happens to be asymmetric, then it is considered transitive.
Tags: arithmetic, asymmetric, negation, opposite, property, quasitransitive, r, relation, transitive, x, y, z
Posted in Arithmetic | No Comments »
Thursday, October 15th, 2009
Complements in Set Theory
Description
A detailed tutorial on complements in set theory. Step by step tutorial including several examples of complements in set theory for reference.
Overview
In set theory, a complement is the opposite of something. It works a little like negation, in that the complement of a set is everything but that set. The way to find this is to subtract the set from its universe, which is a larger set that the set you are taking a complement of belongs to. You can think of your set as a subset of the universe.
Tags: complement, discrete math, elements, Math, negation, opposite, set, set theory, subset, universe
Posted in Discrete Math | No Comments »
Thursday, October 1st, 2009
An Overview of De Morgan’s Laws
Description
A detailed tutorial of De Morgan’s laws. Step by step tutorial including several examples of De Morgan’s laws for reference.
Overview
De Morgan’s laws refer to the logical process of conjunction and disjunction, more commonly known as “and” and “or”. It deals with the negation of entire statements instead of just parts of a statement. De Morgan’s laws state that:
Not (P and Q) = (Not P) or (Not Q)
Not (P or Q) = (Not P) and (Not Q)
In the past, this has been referred to as “complete negation”. It is impossible to solve negations of logical operators without using De Morgan’s laws.
Tags: and, conjunction, contradiction, contrapositive, converse, De Morgan's laws, De Morgan's Rules, discrete math, disjunction, logical operators, Math, negation, not, or, P, Q
Posted in Discrete 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
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 Truth Value Analysis
Description
A detailed tutorial on the introduction to truth value analysis. Step by step tutorial including several examples of the introduction to truth value analysis for reference.
Overview
Truth value analysis is where you get to use negation, conjunction, disjunction, and implication. You first want to start out by making a truth table. A truth table typically uses the letters P and Q, P as the antecedent or first statement and Q as the consequent or second statement. Then you can write in a statement such as “P implies Q” using either conjunction, disjunction, or implication, and declare it as true or false. Remember how to determine if a statement is true or false:
Conjunction: False ^ False = False, everything else is true
Disjunction: True V True = True, everything else is false
Conditional Implication: True => False = False, everything else is true
Biconditional Implication: True <=> True = True, False <=> False = True, everything is is false
Remember that the statements may look long and complicated sometimes, but each part in it can be broken down into something that is true and something that is false to find out if the entire statement is really true or false.
Tags: conjunction, contrapositive, converse, discrete math, disjunction, false, implication, inverse, Math, negation, true, truth value analysis
Posted in Discrete Math | No Comments »
Thursday, September 17th, 2009
Symbols and Translation: Negation
Description
A detailed tutorial on how to translate and symbolize negation. Step by step tutorial including several examples of how to solve problems with negation for reference.
Overview
Negation is a term used in discrete math that refers to the negation or opposite of a statement, which is represented by the ~ symbol before the statement or letter representing the statement. Negation will change a true statement into a false statement, or a false statement into a true statement. When used in statements instead of letters, the implication of a statement is changed instead of the statement itself.
Tags: discrete math, false, Math, negation, opposite, true, ~
Posted in Discrete Math | No Comments »
