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# Mathematical Logic Statements and Notations

## Proposition

A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.

## Predicate

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, ∀ x Y and ∃ 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 ∀.

∀ 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 ∀ 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 ∃.

∃ 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 ∃ 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**

∀ a ∃ b P (x, y) where P (a, b) denotes a + b = 0

∀ a ∀ b ∀ c P (a, b, c) where P (a, b) denotes + (b + c) = (a + b) + c

**Note** − ∀ a ∃ b P (x, y) ≠ ∃ a ∀ b P (x, y)

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