An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements) and a meaning for each predicate symbol. For example, D = {1,2} and P(x): "r> 0" is an interpre- tation for the statement Vr € D, P(x) (in this case, one that happens to make the statement true). For each statement below, provide one interpretation under which the statement is true and another interpretation under which the statement is false- if either case is not possible, explain why clearly and concisely. You may reuse examples if you wish. (a) (b) (c) 3D, Vy D, P(x,y) ⇒ P(y,x) Vr D. Vy D, P(x,y)=P(y,x)]^ Vr D.Vy D.-P(x, y)] D. Q(x)] → [V1 € D, P(x)]
An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements) and a meaning for each predicate symbol. For example, D = {1,2} and P(x): "r> 0" is an interpre- tation for the statement Vr € D, P(x) (in this case, one that happens to make the statement true). For each statement below, provide one interpretation under which the statement is true and another interpretation under which the statement is false- if either case is not possible, explain why clearly and concisely. You may reuse examples if you wish. (a) (b) (c) 3D, Vy D, P(x,y) ⇒ P(y,x) Vr D. Vy D, P(x,y)=P(y,x)]^ Vr D.Vy D.-P(x, y)] D. Q(x)] → [V1 € D, P(x)]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements)
and a meaning for each predicate symbol. For example, D = {1,2} and P(x): "x>0" is an interpre-
tation for the statement Va D, P(x) (in this case, one that happens to make the statement true).
For each statement below, provide one interpretation under which the statement is true and another
interpretation under which the statement is false if either case is not possible, explain why clearly
and concisely. You may reuse examples if you wish.
(a)
(b)
⇒ P(y, x)
3xD, Vy ED, P(x, y)
VrЄD, Vy = D, P(x,y) =
P(y, x)] ^ [Vr ED, Vy Є D, -P(x, y)]
(c)
[3x € D, Q(x)] ⇒ VI € D, P(x)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa61a78e1-5493-43a3-9d06-1e916681d4a2%2F7139ab56-c289-45df-a099-a53cb9748b9b%2Fnbhrl5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements)
and a meaning for each predicate symbol. For example, D = {1,2} and P(x): "x>0" is an interpre-
tation for the statement Va D, P(x) (in this case, one that happens to make the statement true).
For each statement below, provide one interpretation under which the statement is true and another
interpretation under which the statement is false if either case is not possible, explain why clearly
and concisely. You may reuse examples if you wish.
(a)
(b)
⇒ P(y, x)
3xD, Vy ED, P(x, y)
VrЄD, Vy = D, P(x,y) =
P(y, x)] ^ [Vr ED, Vy Є D, -P(x, y)]
(c)
[3x € D, Q(x)] ⇒ VI € D, P(x)]
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