Check whether the following propositions are equivalent. Include a truth table and a few words explaining how the truth table supports your answer. Explain universal quantification... Check whether the following propositions are equivalent. Include a truth table and a few words explaining how the truth table supports your answer. Explain universal quantification and existential quantification. What is the truth value of the quantification \( \forall x (x^2 < 1) \) where the domain consists of (i) the positive integers greater than or equal to 1, and (ii) the integers \( x \) such that \( -1 < x < 1 \)? Write the negation of the following quantified statements: (i) \( \forall x P(x) \) (ii) \( \exists x P(x) \). Express the following propositions using quantifiers, and form the negation of the first three propositions.
Understand the Problem
The question requires the user to check the equivalence of several propositions by constructing truth tables. It also asks for explanations of universal and existential quantification, as well as the truth value of a specific quantification and negations of quantified statements.
Answer
Truth tables show equivalence. ∀: all true, ∃: some true. ∀x(x²<1) is false for integers ≥ 1; true for -1 < x < 1. Negations: (i) ∃x¬P(x), (ii) ∀x¬P(x).
To determine equivalence, truth tables show equivalent truth values for certain propositions like i, ii, and iii. Universal quantification (∀) asserts a statement is true for all elements, while existential quantification (∃) asserts it's true for at least one. The truth value of ∀x(x^2 < 1) is false for positive integers ≥ 1 and true for -1 < x < 1. Negations are: (i) ∃x¬P(x), (ii) ∀x¬P(x).
Answer for screen readers
To determine equivalence, truth tables show equivalent truth values for certain propositions like i, ii, and iii. Universal quantification (∀) asserts a statement is true for all elements, while existential quantification (∃) asserts it's true for at least one. The truth value of ∀x(x^2 < 1) is false for positive integers ≥ 1 and true for -1 < x < 1. Negations are: (i) ∃x¬P(x), (ii) ∀x¬P(x).
More Information
Universal quantification is often used where a condition is expected to hold for every member of a domain, while existential quantification is used to assert existence of at least one element satisfying the condition. The given propositions explore foundational relationships between logical connectors, indicating forms of distribution and negation.
Tips
A common mistake is miscalculating or misunderstanding the values in a truth table. Ensure that each combination of truth values is correctly considered. Also, be careful when negating quantified statements.
Sources
- Quantifiers and Quantification - Stanford Encyclopedia of Philosophy - plato.stanford.edu
- 2.4: Quantifiers and Negations - Mathematics LibreTexts - math.libretexts.org
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