# Truth Table

this note ﻿before ﻿reading this!!

Truth Table for Logical Connectives
(where ' p ' and ' q ' are two statements)
1. Negation

 p q ~﻿p ~q T T F F T F F T F T T F F F T T

2.Conjunction

 p q p ᴧ q T T T T F F F T F F F F

﻿
3.Disjunction

 p q p v q T T T T F T F T T F F F

﻿4.Conditional

 p q p ⇒ q T T T T F F F T T F F T

﻿5. Biconditional

 p q p ⇔ q T T T T F F F T T F F T

Tautology
A compound statement which is always true, whatever may be the truth values of its components, is known as a tautology.
EXAMPLE;

 p q p v q ~﻿p (~﻿p) ᴧ q [(~﻿p) ᴧ q] ⇒ (p v q) T T T F F T T F T F F T F T T T T T F F F T F T

A compound statement which is always false, whatever may be the truth values of its components, is known as a tautology.
EXAMPLE;

 p q p v q ~﻿(p v q) ~﻿(p v q) ᴧ q T T T F F T F T F F F T T F F F F F T F

﻿Laws of Logic [Important for Boolean Algebra (Computer)]
Law of Excluded middle :  Only one statement or
~﻿is true.
Law of Tautology             :  The disjunction of a statement and its negation is                                                           a tautology.
Law of Contradiction      :  The conjunction of a statement and its negation is                                                         a contradiction.
Law of Involution             :  The negation of negation of the statement is                                                                   logically equivalent to a given statement.

Law of Syllogism             :   If p→ q and q→ r then p→ r i.e.,
( p →  q)  Λ (q→  r) →  (p →  r)
Law of Contrapositive     :   ( p ⇒  q)   ≡   (∼ q ) ⇒  (∼ p )
﻿ The conditional and its contrapositive are                                                                     logically equivalent.
Law of Inverse                  :  (∼ p)  ⇒ (∼ q)  ≡ q ⇒p
The inverse and the converse of a conditional                                                              are logically equivalent.