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Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas.
- Predicates
- Quantifiers
- Sample Problems – Predicates and Quantifiers
- Unsolved Problems on Predicates and Quantifiers
- Conclusion – Predicates and Quantifiers
A predicate is a statement that contains variables and becomes a proposition when specific values are substituted for those variables. Predicates express properties or relations among objects. Example: P(x) = “x is an even number” When x=2, P(2) is True. When x=3, P(3) is False.
Quantifiers specify the extent to which a predicate is true over a range of elements. The two main types of quantifiers are universal and existential.
Example 1: Let P(x) be the predicate “x > 5” where x is a real number. Example 2: Let Q(x,y) be the predicate “x + y = 10” where x and y are integers. Q(3,7) is true because 3 + 7 = 10 Q(4,5) is false because 4 + 5 ≠ 10 Example 3: Let R(x) be the predicate “x² ≥ 0” where x is a real number. Example 4: Let S(x) be the predicate “x² = 4” where x is a...
1. Let P(x) be the predicate “x² – 1 = 0” where x is a real number. Determine the truth value of ∃x P(x).2. Let Q(x,y) be the predicate “x < y” where x and y are integers. What does ∀x ∃y Q(x,y) mean in words?3. Let R(x) be the predicate “x is even” where x is an integer. Write the statement “All integers are even” using predicate logic.4. Let S(x) be the predicate “x is a mammal” and T(x) be “x can fly” where x is an animal. How would you express “Some mammals can fly” using predicate logic?Predicates and quantifiers are essential tools in mathematical logic, providing a robust framework for expressing and reasoning about properties and relationships among objects. Their applications in engineering and computer science are vast, ranging from database queries and formal verification to artificial intelligence and mathematical proofs.
- 10 min
13.3 Using the existential quantifier. Consider one more variation of Aristotle’s argument. All men are mortal. Something is a man. _____ Something is mortal. This, too, looks like it must be a valid argument. If the first premise is true, then any human being you could find would be mortal. And, the second premise tells us that something is ...
- Craig DeLancey
- 2017
Expressions in predicate logic with a single quantifier can generally be translated into English as either “there exists an element \(x\) of set \(S\) that satisfies \(P(x)\) ” (existential quantifier) or “every element \(x\) of set \(S\) satisfies \(P(x)\) ” (universal quantifier). However, as the kinds of data we work with grow more complex, we will often find situations where we ...
This quantifier is known as the unique existence quantifier. For example, with the propositional function P(x): x is a cat, if we want to say that all individuals in the universe of discourse are cats, we use the universal quantifier: ∀x: x is a cat. This is read as "for all x, it is true that x is a cat" or in everyday language, "all x are ...
Apr 17, 2022 · Table 2.4: Properties of Quantifiers. In effect, the table indicates that the universally quantified statement is true provided that the truth set of the predicate equals the universal set, and the existentially quantified statement is true provided that the truth set of the predicate contains at least one element.
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The keyword “whenever” suggests that we should use a universal quantifier. \[\forall x,y\,(x\mbox{ is rational} \wedge y\mbox{ is irrational} \Rightarrow x+y\mbox{ is irrational}). \nonumber\] It can also be written as \[\forall x\in\mathbb{Q}\,\forall y\notin\mathbb{Q}\, (x+y\mbox{ is irrational}). \nonumber\] Although this form looks complicated and seems difficult to understand ...
