Jun 25, 2021

Tautology , contradiction, Contingency Definition

 


Define Tautology: -

A Tautology is a formula that is always TRUE for every value of its propositional variables.

Tautology Example: -

Prove: [(A→B) ^A]→B is a tautology

A

B

A→B

(A→B)^A

[(A→B) ^A]→B

TRUE

TRUE

TRUE

TRUE

TRUE

TRUE

FALSE

FALSE

FALSE

TRUE

FALSE

TRUE

TRUE

FALSE

TRUE

FALSE

TRUE

TRUE

FALSE

TRUE


Hene, it’s Tautology

Define Contradiction:-

A contradiction is a formula which is always False for every value its propositional variables.

Contradiction Example :-

prove (A^B)[(¬A) ^ (¬B)] is contradiction.

A

B

A^B

(¬A)

(¬B)

(¬A) ^(¬B)

(A^B)^[(¬A)^(¬B)]

TRUE

TRUE

TRUE

FALSE

FALSE

FALSE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

FALSE

FALSE

TRUE

TRUE

TRUE

FALSE

FALSE

FALSE

FALSE

FALSE

FALSE

TRUE

TRUE

TRUE

FALSE

Hence, it’s contradiction.

Define Contingency: -

A contingency is a formula that has booth some true and false values for every value of irs propositional variables

Contingency Example:-

Prove that (A^B ^ (¬A)

A

B

(A^B)

(¬A)

(A^B)^(¬A)

TRUE

TRUE

TRUE

FALSE

FALSE

TRUE

FALSE

TRUE

FALSE

FALSE

FALSE

TRUE

TRUE

TRUE

TRUE

FALSE

FALSE

FALSE

TRUE

FALSE

Hence, it’s contingency.

PROPOSITIONAL EQUIVALENCE: -

Two statement X or Y are logically equivalent if any of the following two conditions hold-

  • The truth table of each statement has the same truth value.

  • The biconditional statement XY is a tautology.

Example: -

Prove: - ¬(A^B) and [(¬A) ^(¬B)] are equivalents.

Testing by the first method (matching truth table):-

A

B

A^B

¬(A^B)

¬A

¬B

[(¬A)^(¬B)]

TRUE

TRUE

TRUE

FALSE

FALSE

FALSE

FALSE

TRUE

FALSE

TRUE

FALSE

FALSE

TRUE

FALSE

FALSE

TRUE

TRUE

FALSE

TRUE

FALSE

FALSE

FALSE

FALSE

FALSE

TRUE

TRUE

TRUE

TRUE

Hence, the statements are equivalent.

Testing by 2nd method (bi- conditionally)

A

B

¬(A^B)

[(¬A) ^(¬B)]

¬(A^B)⇔[(¬A)^(¬B)]

TRUE

TRUE

TRUE

FALSE

TRUE

TRUE

FALSE

TRUE

FALSE

TRUE

FALSE

TRUE

TRUE

FALSE

TRUE

FALSE

FALSE

FALSE

TRUE

TRUE


Implication /if-then (→) is also called a conditional statement is has two parts

  1. Hypothesis (p)

  2. Conclusion (q)


And it is denoted by p→q

Example:- “if you do your homework, you will not be punished”.

Here, “you do your homework” is the hypothesis(p)

“and you will not be punished” is the conclusion (q)

Inverse:-

An inverse is a conditional statement is the negation of both the Hypothesis and the conclusion if the statement: “If P, then q” Inverse will be “if not then not Q” Thuse the Inverse of p→q is ¬ p→¬q.

Example: -

The Inverse of “If you do your homework, you will not be punished”; is “ if do not your homework you will be punished”.

Converse:-

The converse is the conditional statement is computed by interchanging the  hypothesis and conclusion, if  the statement “if P then q” the converse will be:if a than p” (p→q then q→p) mathematically represented as

Example:

The converse of” if you your homework \, you will not be punished” is “if you will not be punished, you do your homework”.

Contra- Positive: -

The contrapositive of the conditional is computed by interchanging the hypothesis and conclusion of the inverse statement, if the statement is “if p, then q contrapositive is “if not q , then not p” mathematically it represented by p→q is( ¬q→¬p)

 Example:-

“If you do your homework; you will not be punished”

“If you are punished, you did not do your homework”.

Duality Principle:-

Duality principle states that for any true statements, the dual statement, the dual statement obtained by interchanging unions into the intersection (and vice versa )and interchanging universal set into the null set (and vice versa0 If Dual of any statement is the statement itself, it is said the dual statement.

Example:-

The dual of (A ∩B)∪C is (A∪B)∩C

Normal Form: -

We can convert any proposition into two normal forms

  1. Conjunctive Normal Form

  2. Disjunctive Normal Form

Conjunctive Normal Form: -

A compound statement is in conjunctive normal form if it is obtained by operating AND among variables (negation of variables included) connected with ORS. In terms of set operations, It is a compound statement by intersection among variables connected with unions.

Example:-

(A∨B) ^(A∨C) ^(B∨C∨D)

(P∪Q)∩(Q∪R)

Disjunctive Normal Form (DNF): -

A compound statements is in disjunctive normal form if is obtained by operating OR among variables (Negation of Variables included) connected with ANDS in term of set operations, it is a compound statement by union among variables connected with intersection.

Example:-

(A^B)∨(a^ c) ∨(A ^b ^C)

(P∩Q)∪(Q∩R)


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