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the example will be helpful. Thank you.
Give an example of a predicate P(x, y) such that the following two statements are logically...
Example 1.25. The following pairs of statements are logically equivalent. (a) "P(x) is true for all x" 1. (b) "There is no x for which P(x) is not true." (a) "There is some x for which P(x) is true." 2 (b) "It is not the case that P(x) fails for all x." Journal of Inquiry-Based Learning in Mathem Ron Taylor
Example 1.25. The following pairs of statements are logically equivalent. (a) "P(x) is true for all x" 1. (b) "There...
Problem 12.1: Let p and be logical statements. By using a truth table determine if the following compound statements are logically equivalent. Show work! Circle one: A: The statements are equivalent. B: The statements are not equivalent. Problem 12.2: Let P, Q, and be be logical statements. By using a truth table determine if the following compound statements are logically equivalent. Show work! Circle one: A: The statements are equivalent. B: The statements are not equivalent.
question 5.20. Let P(x, y) be the predicate x + y = 10 where x and y are any real numbers. Which of the following statements are true? (a) (∀x)(∃y), P(x, y). (b) (∃y)(∀x), P(x, y)
Let P(X) be the predicate " is a dragon." Let Q(x) be the predicate "x breathes fire.” Let R(x,y) be the predicate "x and y are the same object.” Let S be an arbitrary nonempty set. Rewrite the following English statements in symbolic no- tation using predicates P, Q, R, universal and existential quanti- fiers, and any variables you want. i. There are no dragons in S. ii. Not everything in S is a dragon. iii. There is at least...
4. Use truth tables to determine whether the following two statements are logically equivalent. (P+Q)^(~Q) and ~ (PVQ)
6. Consider the predicates M(x), F(x), and P(x, y) in a domain of people. The predicate M(x) states of a person that he is male, the predicate F(x) states of a person that she is female, the predicate P(z, y) states that x is the parent of y. Write the following queries in the Predicate Logic. (a) Find the people who are mothers (b) Find the people who do not have an uncle.
Give an example of a propositional function P(x,y) such that the statement ∃!x∃!y P(x,y) is true but the statement ∃!y∃!x P(x,y) is false.
4. (8 Points) Using a truth table, prove the following statements are logically equivalent. Be sure to include an explanation of how your truth table demonstrates this conclusion. -(X VY)= -X A-Y
a) Make a table showing the truth value of the predicate p(x, y) = xy ≥ 0 for all possible values of x, y ∈ {−1, 0, 1}. (1 mark) b) Is ∀x∃y p(x, y) true? c) Is ∃x∀y p(x, y) true?
How do you show the following propositions are logically equivalent? (a) [(p → q) → r] ⊕ (p ∧ q ∧ r) and (p ∨ r) ⊕ (p ∧ q) (b) ¬∃x {P(x) → ∃y [Q(x, y) ⊕ R(x, y)] } and (∀x P(x)) ∧ [∀x ∀y(Q(x, y) ↔ R(x, y))] (c) Does [(p → q) ∧ (q → r)] → r implies (p → r) → r?