Consider the open sentence P(x, y,...

Consider the open sentence $$ P(x, y, z):(x-1)^2+(y-2)^2+(z-2)^2>0 . $$ where the domain of each of the variables $x, y$ and $z$ is $\mathbf{R}$. (a) Express the quantified statement $\forall x \in \mathbf{R}, \forall y \in \mathbf{R}, \forall z \in \mathbf{R}, P(x, y, z)$ in words. (b) Is the quantified statement in (a) true or false? Explain. (c) Express the negation of the quantified statement in (a) in symbols. (d) Express the negation of the quantified statement in (a) in words. (e) Is the negation of the quantified statement in (a) true or false? Explain.

Consider the open sentence
$$
P(x, y, z):(x-1)^2+(y-2)^2+(z-2)^2>0 .
$$
where the domain of each of the variables $x, y$ and $z$ is $\mathbf{R}$.
(a) Express the quantified statement $\forall x \in \mathbf{R}, \forall y \in \mathbf{R}, \forall z \in \mathbf{R}, P(x, y, z)$ in words.
(b) Is the quantified statement in (a) true or false? Explain.
(c) Express the negation of the quantified statement in (a) in symbols.
(d) Express the negation of the quantified statement in (a) in words.
(e) Is the negation of the quantified statement in (a) true or false? Explain.

Key Concepts

Translating Between Symbolic Logic and Plain Language

This concept involves converting logical expressions with quantifiers and predicates into verbal statements, and vice versa. Being able to express statements like 'for all' or 'there exists' in plain language helps in understanding the underlying meaning of logical formulas and in communicating mathematical ideas clearly.

Negation of Quantified Statements

Negating a quantified statement involves appropriately switching the quantifiers and negating the predicate. When a universally quantified statement (for all elements, a property holds) is negated, it becomes an existential statement (there exists at least one element for which the property does not hold), and vice versa. This transformation is crucial for proofs involving contrapositions and counterexamples.

Truth Value Analysis

Analyzing the truth value of quantified statements is essential to determine whether the statement holds for every element in the domain (for universal statements) or whether there is at least one counterexample (for existential statements). This process often involves identifying specific cases or counterexamples that either support or refute the general claim.

Open Sentence

An open sentence is a mathematical expression that contains free variables and becomes a full statement only when specific values are assigned to these variables. It serves as a predicate that can be true or false depending on the input values, and is fundamental in logic and predicate calculus.

Universal Quantification

Universal quantification is a logical construct that uses the symbol ?, meaning 'for all.' It asserts that a particular property or statement holds true for every element in a specified domain. This concept is central to forming general statements in mathematics and logic.

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