Math Problem Statement
Solution
The statement given is:
This can be translated as: "For all integers , " is true.
Analysis:
- means "for all " in the set of integers .
- The inequality simplifies to .
This means that the statement is asserting that for all integers, . However, this is not true, because integers like do not satisfy .
Conclusion:
The given statement is false, since there are integers (such as 0 and all negative integers) that do not satisfy the condition .
Would you like further details or explanations on this? Here are some questions to expand your understanding:
- How do we check the validity of a statement with quantifiers?
- Can the inequality be true for a restricted set of integers?
- What would the statement look like if we replaced (for all) with (there exists)?
- How would the truth of the statement change if were restricted to positive integers?
- What are the different types of quantifiers used in mathematical logic?
Tip: Quantifiers like (for all) and (there exists) are key tools in logic and set theory to express properties of elements in a set.
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Math Problem Analysis
Mathematical Concepts
Logic
Set Theory
Inequalities
Formulas
x + 1 ≥ 2
∀x ∈ ℤ
Theorems
Quantifier Logic
Properties of Inequalities
Suitable Grade Level
Grades 10-12
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