LOGICAL OPERATORS

logical operators

LOGICAL OPERATORS:

  Logical operators are the operator which connect two or more proposition by checking logic between them and form a new proposition.

  Some basic logical operations are and the form which easy to understand.

       1.       Conjunction operator

       2.       Disjunction  operator

       3.       Negation operator

       4.       Exclusive OR operator

CONJUNCTION OPERATOR:

  Conjunction operator do the same work that AND operator do, in proposition AND operator named as conjunction operator. For understanding this operator let p and q are two propositions. The conjunction of p and q denoted by p Ù q. the conjunction p Ù q is true when both p and q are true otherwise false at any other conditions. For the result become true all the given proposition/condition must be true.

Truth table for conjunction operator denoted by

Conjunction operator

p	q	p Ù q
T	T	T
T	F	F
F	T	F
F	F	F

DISJUNCTION OPERATOR:

  Disjunction operator do the same work that OR operator do, in proposition OR operator named as disjunction operator. For understanding this operator let p and q are two propositions. The disjunction of p or q denoted by p Ú q. the disjunction p Ú q is false when both p and q are false otherwise true at any other conditions. For the result become true any of the one given proposition/condition must be true.

Truth table for disjunction operator denoted by

disjunction operator

p	q	p Ú q
T	T	T
T	F	T
F	T	T
F	F	F

NEGATION OPERATOR:

   Negation operator change the nature of proposition if the proposition is true if we apply negation operator then the result make false and vice versa. Negation does the same work as NOT operator does. Let p is a proposition then the Ø p truth table for negation operator will be,

P	Ø p
T	F
F	T

Exclusive OR:

  Let  p and q are two proposition. The Exclusive OR p and q are denoted by p Å q is the proposition that is true when exactly one of the p and q is true and false otherwise. It mean p Å q is true if only one proposition is true otherwise if both/all condition true then the result will be false so only one condition/proposition should true.

Truth table for Exclusive OR p Å q.

Exclusive OR operator

p	q	p Å q
T	T	F
T	F	T
F	T	T
F	F	F

  IMPLICATION OPERATOR:

  Let p and q are two propositions the implication of p and represented as p ® q. p ® q is the proposition is false when p is true and q is false, otherwise true at any other condition.

Truth table for implication p®q, where p is hypothesis and q is conclusion.

p	q	p®q
T	T	T
T	F	F
F	T	T
F	F	T

For p ® q:

Ø  If p, then q’

Ø  If p, q

Ø  P implies q

Ø  P only if q

Ø  P is sufficient for q

For p ® q:

Ø  q if p

Ø  q whenever p

Ø  q is necessary for  p

Example:

  P. It is sunny day

 q. we will go to the beach

p ® q will be:

 If it is sunny day, then we will go to the beach. (= if p, then q)

CONVERSE:

 The proposition q ® p is called converse of p ® q.

CONTRAPOSITIVE:

 The contrapositive of p ® q is the proposition Ø q ® Ø p

Example:

 If today is Monday, then I have to go to attend a meeting today.

Converse: If I have to go to attend a meeting today, then today is Monday.

Contrapositive: If I have not to go to attend a meeting today, then today is not Monday.

BICONDITIONAL OPERATOR:

 The last logical operator of proposition, which is p « q, if p and q are propositions. p « q true when if the p and q are same truth value. Means all condition should true or false then the result will true otherwise false.

Truth table for Biconditional operator p « q is,

p	q	p « q
T	T	T
T	F	F
F	T	F
F	F	T

For p « q,

Ø  p, if and only if q (p if if q)

Ø  p is necessary and sufficient for q

Ø  if p, then q is conversely.

Example:

 You can take the flight if and only if you buy a ticket. 

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