Universal and Existential Quantifiers

In this class, We discuss Universal and Existential Quantifiers.

The reader should have prior knowledge of predicate logic basics. Click Here.

We take examples and understand universal and existential quantifiers.

Example:

Apple is red.

The above statement simply says the apple is red.

Every apple is red.

The above statement says every apple is red.

The “Every apple is red” statement quantifies every apple.

Quantify means to what extent the statement is correct.

We have two types of quantifiers.

1) Universal quantifier

2) Existential quantifier

Universal quantifier:

All men will die

Every apple is red.

The universal quantifier is applicable to all.

First, we need to understand the universe of discourse.

The universe of discourse contains all the elements.

All men, women, living beings, non-living beings, etc, belong to the universe of discourse.

Based on the universe of discourse, we need to write the statements.

All men will die is written below.

M: is man

D: will die

(∀x)(M(x) -> D(x))

for all x, if x is a man, then he will die.

If x is a man because our universe of discourse contains all.

The statement is true: if he is a man, then he will die.

The symbol for ” for all x” ∀x or (x).

Take the universe of discourse that contains only men.

We write as (∀x)(D(x)) because our universe contains only men.

2) Existential quantifier

Some men are clever.

(∃x)(M(x) ∧ C(x))

There exists at least one man who is clever

x is a man, and x is clever. The equation M(x) ∧ C(x) will be true.

If at least one man is found, then (∃x)(M(x) ∧ C(x)) will be true.