Logical Operations And Logical Connectivity

Negation (NOT): Given any proposition P, another proposition, called negation of P, can be formed by writing “It is not the case that…….. or”. “It is false
that…….” before P or, if possible, by inserting in P the word “not”. Symbolically ¬ P or ~ P read “not P” denotes the negation of P. the truth value of ¬ P depends on the truth value of P. If P is true then ¬ P is false and if P is false then ¬ P is true. The truth table for Negation is as follows :

Example 5 :
Let P : 6 is a factor of 12.
Then Q = ¬ P : 4 is not a factor of 12. Here P is true & ¬ P is false.

Conditional or Implication : (If……then): If two statements are combined by using the logical connective ‘if….then’ then the resulting statement is called a conditional statement.
If P and Q are two statements forming the implication “if P then Q” then we denotes this implication P→Q. In the implication P→Q, P is called antecedent or hypothesis Q is called consequent or conclusion.
The statement P→Q is true in all cases except when P is true and Q is false.

The truth table for implication is as follows.

Since conditional statement play an essential role in mathematical reasoning a variety of terminology is used to express P→Q.
i) If P then Q
ii) P implies Q
iii) P only if Q
iv) Q if P
v) P is sufficient condition for Q
vi) Q when P
vii) Q is necessary for P
viii) Q follows from P
ix) if P, Q
x) Q unless ¬ P
Converse, Inverse and Contrapositive of a conditional statement : We can form some new conditional statements starting with a conditional statements related conditional statements that occur so often that they have special names → converse, contrapositive & Inverse. Starting with a conditional statement P→Q that occur so often that they have special names.

3. Inverse : If P→Q is an implication then ¬ P → ¬ Q is called its
inverse.
Example 6 : Let P : You are good in Mathematics.
Q : You are good in Logic.
Then, P→Q: If you are good in Mathematics then you are good in Logic.
1) Converse : (Q→P)
If you are good in Logic then you are good in Mathematics.
2) Contrapositive : ¬ P → ¬ Q
If you are not good in Logic then you are not good in Mathematics.
3) Inverse : (¬ P → ¬ Q)
If you are not good in Mathematics then you are not good in Logic.