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# Discrete Mathematics - Predicate Logic

**Predicate Logic** deals with predicates, which are propositions containing variables.

## Predicate Logic – Definition

A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.

The following are some examples of predicates −

- 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"

## Well Formed Formula

Well Formed Formula (wff) is a predicate holding any of the following −

All propositional constants and propositional variables are wffs

If x is a variable and Y is a wff, $\forall x Y$ and $\exists x Y$ are also wff

Truth value and false values are wffs

Each atomic formula is a wff

All connectives connecting wffs are wffs

## Quantifiers

The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier.

### Universal Quantifier

Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol $\forall$.

$\forall x P(x)$ is read as for every value of x, P(x) is true.

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

### Existential Quantifier

Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol $\exists $.

$\exists x P(x)$ is read as for some values of x, P(x) is true.

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

### Nested Quantifiers

If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier.

**Example**

$\forall\ a\: \exists b\: P (x, y)$ where $P (a, b)$ denotes $a + b = 0$

$\forall\ a\: \forall\: b\: \forall\: c\: P (a, b, c)$ where $P (a, b)$ denotes $a + (b + c) = (a + b) + c$

**Note** − $\forall\: a\: \exists b\: P (x, y) \ne \exists a\: \forall b\: P (x, y)$