Transforming predicate form to quantifiers

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In summary, transforming predicate form to quantifiers is done to simplify and clarify logical statements. The two basic types of quantifiers used are universal and existential quantifiers. To transform a predicate form with multiple variables, the subject and predicate are identified and quantifiers are used to express their relationship. The main difference between universal and existential quantifiers is the scope of their statements. Some common mistakes to avoid when transforming predicate form to quantifiers include using the wrong quantifier, not specifying the domain, and using incorrect variables. It is crucial to carefully consider the original statement and accurately represent it using quantifiers.
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MarcL
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Homework Statement


Write the following statements in predicate form, using logical operators ^,∨, (NOT - negation but don't know where the symbol is :/) , and quantifiers ∀,∃. Below ℤ+ denotes all positive integers {1,2,3,...}.

I need help with this first statement:
For any x, y ∈ ℤ+ the equation x2 + y2 - z = 0 has a solution z
∈ ℤ+

Homework Equations


The fact that ∀ is universal and ∃ is existential.

The Attempt at a Solution



I can't seem to figure out where to start... I just got ∀x∀y ( x2 + y2 = z) which seems wrong to me. any help?
 
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  • #2
"Has a solution" means "there exists a solution".
 

What is the purpose of transforming predicate form to quantifiers?

The purpose of transforming predicate form to quantifiers is to make logical statements more concise and easier to understand. By using quantifiers, we can express statements that would otherwise require multiple predicate symbols and operators.

What are the basic types of quantifiers used in transforming predicate form?

The two basic types of quantifiers used in transforming predicate form are universal quantifiers and existential quantifiers. Universal quantifiers (∀) are used to express statements that are true for all values in a given domain, while existential quantifiers (∃) are used to express statements that are true for at least one value in a given domain.

How do you transform a predicate form with multiple variables to quantifiers?

To transform a predicate form with multiple variables to quantifiers, we first identify the subject and predicate of the statement. Then, we use quantifiers (∀ or ∃) to express the relationship between the subject and predicate. For example, the statement "All cats are mammals" can be transformed to ∀x (Cat(x) → Mammal(x)), where x represents any cat.

What is the difference between universal and existential quantifiers?

The main difference between universal and existential quantifiers is the scope of their statements. Universal quantifiers make statements that are true for all values in a given domain, while existential quantifiers make statements that are true for at least one value in a given domain. In other words, universal quantifiers are inclusive, while existential quantifiers are more specific.

What are some common mistakes to avoid when transforming predicate form to quantifiers?

Some common mistakes to avoid when transforming predicate form to quantifiers include using the wrong quantifier (∀ vs ∃), not specifying the domain of the quantifier, and using incorrect variables in the quantified statement. It is important to carefully consider the logical structure of the original statement and accurately represent it using quantifiers.

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