Discrete Mathematics Rules of Interface - Discrete Mathematics

What are the Rules of Interface for Discrete Mathematics?

Rules of Interface are used for deducing the new statements from the true statements.

In order to determine the truth values of the mathematical statements the valid arguments that are used are proofs and for logical proofs, mathematical logic is used.

A sequence of statements is known as argument and the last statement of the sequence is conclusion and the preceding statements are known as premises or hypothesis. Prior to conclusion, the symbol “∴” is used. When the conclusion is the outcome of the truth values of the premises, the argument is considered to be a valid argument.

From the already provided statements, valid arguments can be constructed by using the templates and guidelines provided by Rules of Interface.

Table of Rules of Inference

Rule of Inference
Name
Rule of Inference
Name
Addition
Disjunctive Syllogism
Conjunction
Hypothetical Syllogism
Simplification
Constructive Dilemma
Modus Ponens
Destructive Dilemma
Modus Tollens

Addition

For the premise P, Addition Rule is used for deriving P∨Q. Addition

AdditionPPQ

Example

Let P be the proposition, “He studies very hard” is true

Therefore − "Either he studies very hard Or he is a very bad student." Here Q is the proposition “he is a very bad student”.

Conjunction

For two premises P and Q, Conjunction rule is used to derive P∧Q.

ConjunctionPQPQ

Example

Let P − “He studies very hard”

Let Q − “He is the best boy in the class”

Therefore − "He studies very hard and he is the best boy in the class"

Simplification

For the premise P∧Q, p can be derived by using the simplification rule.

SimplificationPQP

Example

"He studies very hard and he is the best boy in the class", P∧Q

Therefore − "He studies very hard"

Modus Ponens

For the premises P and P→Q, Q can be derived by using Modus Ponens.

Modus PonensPQPQ

Example

"If you have a password, then you can log on to facebook", P→Q

"You have a password", P

Therefore − "You can log on to facebook"

Modus Tollens

For the premises P→Q and ¬Q, Modus Tollens is used to derive ¬P.

Modus TollensPQ¬Q¬P

Example

"If you have a password, then you can log on to facebook", P→Q

"You cannot log on to facebook", ¬Q

Therefore − "You do not have a password”

Disjunctive Syllogism

For two premises ¬P and P∨Q, disjunctive syllogism is used to derive Q.

Disjunctive¬PPQQ

Example

"The ice cream is not vanilla flavored", ¬P

"The ice cream is either vanilla flavored or chocolate flavored", P∨Q

Therefore − "The ice cream is chocolate flavored”

Hypothetical Syllogism

For the premises P→Q and Q→R, Hypothetical Syllogism is used to derive P→R.

HypotheticalPQQRPR

Example

"If it rains, I shall not go to school”, P→Q

"If I don't go to school, I won't need to do homework", Q→R

Therefore − "If it rains, I won't need to do homework"

Constructive Dilemma

For the two premises (P→Q)∧(R→S) and P∨R, constructive dilemma is used to derive Q∨S.

(PQ)(RS)PRQS

(PQ)(RS)PRQS

Constructive Dilemma

(PQ)(RS)PRQS

Example

“If it rains, I will take a leave”, (P→Q)

“If it is hot outside, I will go for a shower”, (R→S)

“Either it will rain or it is hot outside”, P∨R

Therefore − "I will take a leave or I will go for a shower"

Destructive Dilemma

For the two premises P→Q)∧(R→S) and ¬Q∨¬S, destructive dilemma is used for deriving ¬P∨¬R.

Destructive Dilemma

Example

“If it rains, I will take a leave”, (P→Q)

“If it is hot outside, I will go for a shower”, (R→S)

“Either I will not take a leave or I will not go for a shower”, ¬Q∨¬S

Therefore − "Either it does not rain or it is not hot outside"

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