Predicate Logic and Proofs

In summary, the given statement is translated to predicate logic as: for all positive real numbers x and positive integers z, there exists a positive real number y such that yz2 > xz + 10. This can be proved using a direct approach by defining the predicate Q (x, y, z) = yz2 > xz + 10 and showing that for any given positive real number x and positive integer z, there exists a positive real number y that satisfies Q.
  • #1
SolarMidnite
22
0

Homework Statement



No matter what positive real number x we choose, there exists some positive real number y
such that yz2 > xz + 10 for every positive integer z.

Translate the above statement to predicate logic and prove it using a direct approach.

Homework Equations



I don't believe that there are relevant equations for this problem.

The Attempt at a Solution



Let Q (x, y, z) = yz2 > xz + 10

[itex]\forall[/itex]x ∈ ℝ+ [itex]\exists[/itex]y ∈ ℝ+ [itex]\forall[/itex]z ∈ [itex]Z[/itex]+ Q(x, y, z)

Before I attempted to prove the theorem, I wanted to make sure that I got the predicate logic translation right. I don't think that the above translation is right, but I hope I'm on the right track. I've never translated into predicate logic with 3 variables. It's usually just x and y, so should it be (x, y, z)? Also, does [itex]\forall[/itex]z ∈ [itex]Z[/itex]+ come after Q (x, y, z) since it does in the statement?
 
Physics news on Phys.org
  • #2
SolarMidnite said:

Homework Statement



No matter what positive real number x we choose, there exists some positive real number y
such that yz2 > xz + 10 for every positive integer z.

Translate the above statement to predicate logic and prove it using a direct approach.

Homework Equations



I don't believe that there are relevant equations for this problem.

The Attempt at a Solution



Let Q (x, y, z) = yz2 > xz + 10

[itex]\forall[/itex]x ∈ ℝ+ [itex]\exists[/itex]y ∈ ℝ+ [itex]\forall[/itex]z ∈ [itex]Z[/itex]+ Q(x, y, z)

Before I attempted to prove the theorem, I wanted to make sure that I got the predicate logic translation right. I don't think that the above translation is right, but I hope I'm on the right track. I've never translated into predicate logic with 3 variables. It's usually just x and y, so should it be (x, y, z)? Also, does [itex]\forall[/itex]z ∈ [itex]Z[/itex]+ come after Q (x, y, z) since it does in the statement?
I would write it this way.

[itex]\forall (x ∈ R^+, z ∈ Z^+) \exists y ∈ R^+ \ni Q(x, y, z)[/itex]

In addition to other changes, I also replaced ℝwith R, since ℝis so tiny I can barely tell it's a version of the letter R.

There's a nicer one that you can get with mathbb{R}, as in
[itex]\mathbb{R}[/itex].
 

1. What is predicate logic?

Predicate logic is a formal system of logic that deals with propositions involving predicates, or properties, and quantifiers, which specify the quantity of objects that satisfy a given predicate. It is used to reason about the truth of statements that involve variables and sets of objects, making it a powerful tool in mathematics and computer science.

2. How is predicate logic different from propositional logic?

Unlike propositional logic, which only deals with the truth values of simple statements, predicate logic allows for more complex statements involving variables and quantifiers. This makes it a more expressive and versatile system for formal reasoning.

3. What is a proof in predicate logic?

A proof in predicate logic is a sequence of logically valid steps that demonstrate the truth of a given statement, known as the conclusion. These steps are based on a set of axioms and rules of inference, and they must follow a specific format in order to be considered a valid proof.

4. What is the significance of universal and existential quantifiers in predicate logic?

Universal and existential quantifiers, denoted by ∀ and ∃ respectively, are essential tools in predicate logic for expressing general statements about sets of objects. The universal quantifier (∀) denotes that a statement applies to all objects in a given set, while the existential quantifier (∃) denotes that at least one object in the set satisfies the statement.

5. How is predicate logic used in real-world applications?

Predicate logic has a wide range of applications in various fields, including mathematics, computer science, philosophy, and linguistics. It is used to formalize and reason about complex systems and structures, such as programming languages, natural language semantics, and mathematical theories. It also plays a crucial role in automated theorem proving and formal verification, making it an essential tool in the development of reliable and secure computer systems.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
403
  • Precalculus Mathematics Homework Help
Replies
5
Views
800
  • Precalculus Mathematics Homework Help
Replies
5
Views
951
  • Precalculus Mathematics Homework Help
Replies
6
Views
993
  • Calculus and Beyond Homework Help
Replies
2
Views
590
  • Precalculus Mathematics Homework Help
Replies
20
Views
896
  • Precalculus Mathematics Homework Help
Replies
24
Views
2K
  • Precalculus Mathematics Homework Help
Replies
12
Views
2K
Back
Top