# Discrete Mathematics Predicate Logic - Discrete Mathematics

## What is predicate logic in Discrete Mathematics?

The predictions with respect to propositions are made by Predicate Logic.

## Define Predicate Logic.

The expression of one or more variables are defined on the same specific domain, is defined as predicate. The variables predicate can be made on a proposition, either by assigning a value to the variable or by quantifying the variable.

Some of the examples of predicates are as follows:

• Let E(x, y) denote "x = y"
• Let X(a, b, c) denote "a + b + c = 0"
• Let M(x, y) denote "x is married to y"

## What is Well Formed Formula?

A predicate that holds any one of the following conditions is known as Well Formed Formula (wff).

• All the propositional variables and propositional constants are Wffs.
• If x is a variable and Y is a wff, ∀xY and ∃xY are also wff
• The values of true and false are wffs
• Each of the atomic formula is a wff
• All the connectives that connect wffs are wffs

## What are Quantifiers?

Quantifiers are used for quantifying the variables of prediction. The quantifiers in predicate logic are of two types - Universal Quantifier and Existential Quantifier.

### Universal Quantifier

For every value of the specific variable, the statements within the scope are defined to be true by the universal quantifier and are denoted by ∀.

∀xP(x) is read as for every value of x, P(x) is true.

Example − "Man is mortal" can be transformed into the propositional form ∀xP(x) where P(x) is the predicate which denotes x is mortal and the universe of discourse is all men.

### Existential Quantifier

For some of the values of specific variable, the statements within the scope are defined as true by existential quantifier and are denoted by ∃.

∃xP(x) is read as for some values of x, P(x) is true.

Example − "Some people are dishonest" can be transformed into the propositional form ∃xP(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people.

### Nested Quantifiers

When the quantifier that appears within the scope of another quantifier is used, it is known as nested quantifier.

Example

• ∀ a∃bP(x,y) where P(a,b) denotes a+b=0
• ∀ a∀b∀cP(a,b,c) where P(a,b) denotes a+(b+c)=(a+b)+c

Note ∀a∃bP(x,y)≠∃a∀bP(x,y)