## Introduction

The universal quantifier is denoted by ∀”.

An example of how to interpret it is as follows:

The statement “∀x P(x) is true if P(x) holds for all x in the universe of discourse. On the other hand, if you can find x such that P(x) is false, then the statement “∀x P(x) is false.

The existential quantifier is denoted by ∃.

The statement ∃x P(x) is true if P(x) holds for at least one x in the universe of discourse. On the other hand, you cannot find at least one x such that P(x) is true, then the statement ∃x P(x) is false.

**Example 1**

**Example 2**

Now, let’s solve one of the exercises so you can have an example to follow while you solve more exercises.

## Exercise 9

Let P (x) be the statement “x can speak Russian” and let Q(x) be the statement “x knows the computer language C++.” Express each of these sentences in terms of P (x), Q(x), quantifiers, and logical connectives. The domain for quantifiers consists of all students at your school.

**a) **There is a student at your school who can speak Russian and who knows C++.

**b) **There is a student at your school who can speak Russian but who doesn’t know C++.

**c) **Every student at your school either can speak Russian or knows C++.

**d) **No student at your school can speak Russia nor knows C++.

### Answers

P(x) = “x can speak Russian”

Q(x) = “x knows the computer language C++.”

- ∃x (P(x) ^ Q(x))
- ∃x (P(x) ^ ¬Q(x))
- ∀x (P(x) v Q(x))
- ¬∃x (P(x) ^ Q(x))

Notice that the quantifiers have precedence over all the operators used in propositional calculus. That’s why we have to use parenthesis to specify what is the scope of the quantifier. ∃x (P(x) ^ Q(x)) is not the same than ∃x P(x) ^ Q(x).

Also notice that the following logical equivalences holds for quantifiers:

- ¬∀xP (x) ≡ ∃x ¬P (x)
- ¬∃xQ(x) ≡ ∀x ¬Q(x)

So, the answer to question d can also be written as ∀x ¬ (P(x) ^ Q(x))

**Related topics:**

- Predicate Logic and Quantifiers
- Let P(x) be the statement “x = x
^{2}.” If the domain consists of the integers, what are these truth values? - How to create the truth table for a compound proposition
- 7. Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people