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, ∀xY$\mathrm{\forall }xY$ and ∃xY$\mathrm{\exists }xY$ 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 ∀$\mathrm{\forall }$.

∀xP(x)$\mathrm{\forall }xP(x)$ 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)$\mathrm{\forall }xP(x)$ 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 ∃$\mathrm{\exists }$.

∃xP(x)$\mathrm{\exists }xP(x)$ 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)$\mathrm{\exists }xP(x)$ 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

• ∀ a∃bP(x,y)$\mathrm{\forall }a\mathrm{\exists }bP(x,y)$ where P(a,b)$P(a,b)$ denotes a+b=0$a+b=0$
• ∀ a∀b∀cP(a,b,c)$\mathrm{\forall }a\mathrm{\forall }b\mathrm{\forall }cP(a,b,c)$ where P(a,b)$P(a,b)$ denotes a+(b+c)=(a+b)+c$a+(b+c)=(a+b)+c$

Note − ∀a∃bP(x,y)≠∃a∀bP(x,y)