Predicate Logic ,Quantifiers and Nested quantifiers in Discrete Mathematics|UGC NET Computer Science notes

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In this post,let us understand predicate logic ,  quantifiers and nested quantifier in discrete mathematics in a simple way.

Introduction

First ,let us know the difference between Propositions and Predicates.

Propositions  deal  with declarative statements that is either true or false ,whereas Predicates deal with statements that contain variables and objects.

What is Predicate logic?

 Predicate logic is an extension of propositional logic that includes variables ,objects and quantifiers to represent more complex statements.

Example:

Propositions

Rain scenario:

In Madurai,It is raining.

In Chennai,It is not raining.

We use seperate letters for each statement

Let

P=In  Madurai,it is raining

Q=In Chennai,it is raining

But Chennai is not raining

Given statement:

It is raining in Madurai and it is not raining in Chennai.

Now we write this statement in Propositions form

P ∧ ¬Q

Predicates:

In predicate logic, we use one predicate with multiple variables.

Statement:

It is raining in Madurai  and it is not raining in chennai

R(x) denotes it is raining in city.

R is predicate , m and c is the name of the city.

Now we write the given statement in predicate form

R(m) ∧ ¬R(C)

Explanation:

Look at the above example, In propositions,we use more letters to represent that scenario.But we use one letter with multiple variables to represent that scenario.

Quantifiers:

Quantifiers are symbols used to quantify the variables in predicate logic .

There are two types of Quantifiers.

• Universal Quantifier( ∀)
• Existential Quantifier( ∃)

Universal  Quantifier:

Universal Quantifier means the given statement is true for all value of a variable.

For example:

Rain scenario:

It is raining in all cities.

In Universal Quantifier,we write  ∀x R(x)

Existential  Quantifier:

Existential Quantifier means the given statement is true for some value of a variable.

For example:

Rain scenario:

It is raining in some city

In Existential Quantifier, we write  ∃x R(x).

Symbol.           Meaning

∀.                For all values

∃.                For some values or atleast  one value

Nested Quantifiers:

Nested Quantifiers are quantifier occur within

the scope of another quantifier.

For example

• All Students have some discrete mathematics books.

Student=x, Discrete Mathematics book=y

We write, ∀x∃yP(x,y)

• Some students have all books

We write,∃x∀yP(x,y)

Conclusion:

I hope,this post helps you to understand predicate logic,quantifiers and nested quantifiers using rain scenario example in a simple way.