Mathematical Logic

Mathematical Logic is a way of performing operations in Maths. It is much like functions, but the basic difference is that in functions , we perform an operation over the equation of the y coordinate , while in logic , we perform an operation over the inputs of a circuit.

Logic
Logic is an operation performed over the inputs of a circuit to give an output. There are many such different logical operators in Mathematics and digital electronics. The logic operators included in the syllabus are mentioned below. The number of possible outcomes is given by 2 raised to number of inputs.

AND Logic
In AND logic, the output will be true only when all inputs are true. The output will be false even if one input is false. It is denoted by ^. ex. p^q. Truth table

OR Logic
In OR Logic, the output will be true even if one input is true. The output will be false only if all the inputs are false. It is denoted by v. ex. pvq

NOT Logic
In NOT Logic, the output will be the opposite of input. If the input is true, the output will be false. NOT logic is also referred to as Compliment or negation. It is denoted by ~. ex. ~p. Note : NOT logic has only one input.

Implication
Under implication, the output will be false only if the first input is true and second input is false. in the rest cases, output will always be True. It is denoted by →. ex. p→q. Note : Implication has only two inputs. Truth Table The equation of implication is given as :- p→q≈~pvq TruthTable

Double Implication
Under double implication, the output is true , only when the number of "True" inputs is even. If there are 0 or 2 or 4 .... True values, the output will be true. It is denoted by ←→. ex. p ←→ q.

Logic Circuits
In logic circuits, OR stands for parallel and AND stands for series connections.

Simplification
p^T =p p^F =F pvT =T pvF =p p^~p=F pv~p=T

~(p^q) = ~pv~q ~(pvq) = ~p^~q

Statements
A statement is a declarative sentence which is either true or false. Assertive sentences are statement. Imperative, exclamatory and interrogative sentences are not statements.

Converse, Inverse and Contrapositive
Converse : q ---> p

Inverse : ~p ---> ~q

Contrapositive : ~q ---> ~p

Tautology, Contradiction and Contingency
Tautology : When all outputs are true

Contradiction : When all outputs are false

Contingency : When the outputs are a combination of true and false values

Quantifiers
Universal Quantifier : ∀

Existential Quantifier : ∃

Principle of Duality
Under the principle of duality, the logic signs are simply interchanged. AND becomes OR. OR becomes AND. Tautology becomes contradiction. Contradiction becomes Tautology.

Negation of Compound Statements
(De Morgan's Theorems )

~(pvq) = ~p^~q

~(p^q) = ~pv~q