Quantifiers are expressions that indicate the scope of the term to which they are attached, here predicates. A predicate is a property the subject of the statement can have.
For example, in the statement “the sum of x and y is greater than 5”, the predicate ‘Q’ is- sum is greater than 5,
and the statement can be represented as Q(x, y) where x and y are variables.
The scope of a quantifier or a quantification is the range in the formula that the quantifier engages in.
Types of quantification or scopes:
- Universal(∀) – The predicate is true for all values of x in the domain.
- Existential(∃) – The predicate is true for at least one x in the domain.
To know the scope of a quantifier in a formula, just make use of Parse trees. Two quantifiers are nested if one is within the scope of the other.
∀x ∃y (x+y=5)
Here ‘∃’ (read as-there exists) and ‘∀’ (read as-for all) are quantifiers for variables x and y.
The statement can be represented as-
Q(x) is ∃y P(x, y) Q(x)-the predicate is a function of only x because the quantifier applies only to variable x.
P(x, y) is (x + y = 5)
∀x ∀y ((x> 0)∧(y> 0) → (xy< 0))
For every real number x and y, if x is positive and y is negative, implies xy is negative.
where Q(x) is ∀y P(x, y)
Example to convert a statement into a nested quantifiers formula:
“There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
A statement consists of quantifiers and predicates, split it into it's two constituents.
Here x and y are the pupil and the course and their respective quantifiers are attached in front of them.
Write it down as-
For some x pupil, there exist a course in Discrete Maths such that x has taken y.
∃x ∃y P (x, y), where P (x, y) is "x has taken y".
- Theorem-1: The order of nested existential quantifiers can be changed without changing the meaning of the statement.
- Theorem-2: The order of nested universal quantifiers can be changed without changing the meaning of the statement.
Assume P(x, y) is xy=8,
∃x ∃y P(x, y) domain: integers
There is an integer x for which there is an integer y such that xy = 8,
which is same as-
There is a pair of integers x, y for which xy = 8.
Meaning ∃x ∃y P(x, y) is equivalent to ∃y ∃x P(x, y).
Assume P(x, y) is (xy = yx).
∃x ∃y P(x, y) domain: real numbers
For all real numbers x, for all real numbers y, xy = yx or,
For every pair of real numbers x, y, xy = yx.
again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y).
However, when the nested quantifiers are not same, changing the order changes meaning of statement.
Assume P(x, y, z) is (x + y = z).
∀x ∀y ∃z P(x, y, z) domain: real numbers
For all real numbers x and y there is a real number z such that x + y = z (True)
∀z ∃x ∃y P(x, y, z) domain: real numbers
There is a real number z such that for all real numbers x and y, x + y = z (False)
Negation of nested quantifiers:
To negate a sequence of nested quantifiers, you change each quantifier in the sequence to the other type and then negate the predicate.
So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : ~P(x, y)
“ ∃x at Cornell, x is at least 18 years old.”
To disagree with this, you’re negating the statement by flipping the ∃ to ∀ and then
negating the the predicate:
“ ∀x at Cornell such that x is not at least 18 years old.”
- Mathematics | Predicates and Quantifiers | Set 2
- Mathematics | Predicates and Quantifiers | Set 1
- Mathematics | Probability
- Mathematics | Generalized PnC Set 1
- Mathematics | Generalized PnC Set 2
- Mathematics | Combinatorics Basics
- Mathematics | Introduction to Proofs
- Mathematics | Rules of Inference
- Mathematics | Indefinite Integrals
- Mathematics | Algebraic Structure
- Mathematics | Random Variables
- Mathematics | Lagrange's Mean Value Theorem
- Mathematics | Rolle's Mean Value Theorem
- Mathematics | Conditional Probability
- Mathematics | Introduction of Set theory
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
Improved By : VaibhavRai3