Math Problem Statement

For all integers x, the statement (x + 1 ≥ 2) is true.

Solution

The statement given is:

(xZ)(x+12) is true.(\forall x \in \mathbb{Z}) (x + 1 \geq 2) \text{ is true.}

This can be translated as: "For all integers xx, x+12x + 1 \geq 2" is true.

Analysis:

  • xZ\forall x \in \mathbb{Z} means "for all xx" in the set of integers Z\mathbb{Z}.
  • The inequality x+12x + 1 \geq 2 simplifies to x1x \geq 1.

This means that the statement is asserting that for all integers, x1x \geq 1. However, this is not true, because integers like x=0,1,2,x = 0, -1, -2, \dots do not satisfy x1x \geq 1.

Conclusion:

The given statement (xZ)(x+12)(\forall x \in \mathbb{Z}) (x + 1 \geq 2) is false, since there are integers (such as 0 and all negative integers) that do not satisfy the condition x+12x + 1 \geq 2.

Would you like further details or explanations on this? Here are some questions to expand your understanding:

  1. How do we check the validity of a statement with quantifiers?
  2. Can the inequality x+12x + 1 \geq 2 be true for a restricted set of integers?
  3. What would the statement look like if we replaced \forall (for all) with \exists (there exists)?
  4. How would the truth of the statement change if xx were restricted to positive integers?
  5. What are the different types of quantifiers used in mathematical logic?

Tip: Quantifiers like \forall (for all) and \exists (there exists) are key tools in logic and set theory to express properties of elements in a set.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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