Properties of Sets
Truth Table
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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 |
Contradiction
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 p or ~p 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.
