Day 33: Mathematical Logic

Hello Dear Students,
 Hope you all are doing good.

Aaj hum mathematical logic ke baare mein study karenge.

Let's get started...

Mathematical logic means simply jo logic se hum kisi problem ko solve karte hain. It is basically combination of both, mathematics, linguistics and theoretical computer science. Logic is basically of 2 types which are -
  1. Propositional Logic
  2. Predicate Logic

  • Propositional logic is fact that the solution can be even true or false but not both. 
  • Propositional logic mein values only true hongi or false, but not both.
  • We represent true as 1 and false as 0.
  • For example, 3+3=6, it is a propositional logic with value True(1). 
  • The logical operations on propositions are as follows-
1. Conjunction - Conjunction is described as "and". If 2 propositions are there, suppose p and q, then it is denoted as p^q

Truth Table for p^q

2. Disjunction - Disjunction is described as "or". If 2 propositions are there, suppose p and q, then it is denoted as p v q.

Truth Table for p v q
 pp v q 

3. Negation - Negation is described as "not". If a propositions are there, suppose p, then it is denoted as ~p. 

Truth Table for ~p

4. Implications - Implication is described as "implied to",basically statement if then is called an implication. Implies mein ek value premise or hypothesis hoti hai and doosri value conclusion or consequence hoti hai. For example, If it is breakfast, then this is morning yet. Isme if it is breakfast is premise and then this is morning yet is conclusion. If 2 propositions are there suppose p and q, then it is denoted as p=>q.

Truth Table for p=>q

In above table, we can say that p is premise and q is conclusion.

If p=>q is implication then q=>p is called as converse and ~q=>~p then it is called contrapositive. 

Propositions Types 

Proposition means either true or false but not both. So types or propositions are-
  1. Tautology - A proposition which is always true is called Tautology. 
  2. Contradiction - A proposition which is always false is called Contradiction.
  3. Contingency - A proposition that is neither a tautology nor a contradiction is called a contingency.

  • Predicate logic is also known as First Order Logic and First Order Predicate Calculus. 
  • Predicate logic uses Quantified Variables,  objects and relations. 
  • For example, Ram is a student =  Ram(Student).
Quantifiers- Predicates ke variables ko quantify karne ke liye quantifiers use kiye jate hain. Quantify means basically measure or quantity. There are 2 types of quantifiers basically-
  1. Universal Quantifier - It means the statements are true for all variables. It is denoted by .For example, all boys like cricket, x;boys(x)->like(x,cricket).
  2. Existential Quantifier -  It means the statements are true for some of the variables and not all. It is denoted by . For example, some boys like cricket, x;boys(x)^like(x,cricket). 

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