Show that the two quantified statements in each problem are not logically equivalent by filling in a table so that, for the domain (a, b, c), the values of the predicate P you select for the table causes one of the statements to be true and the other to be false. For example, the table below shows that vx vy P(x, y) and 3x3y P(x, y) are not logically equivalent because for the given values of the predicate P, vx vy P(x, y) is false and 3x 3y P(x, y) is true. P a b C a T T T b T F T T T F (a) vx³y P(x, y) and 3x vy P(x, y) (b) vx³y ((x+y) P(x, y)) and vx 3y P(x, y) (c) 3x³ (P(x, y) AP(y, x)) and 3x3 P(x, y)
Show that the two quantified statements in each problem are not logically equivalent by filling in a table so that, for the domain (a, b, c), the values of the predicate P you select for the table causes one of the statements to be true and the other to be false. For example, the table below shows that vx vy P(x, y) and 3x3y P(x, y) are not logically equivalent because for the given values of the predicate P, vx vy P(x, y) is false and 3x 3y P(x, y) is true. P a b C a T T T b T F T T T F (a) vx³y P(x, y) and 3x vy P(x, y) (b) vx³y ((x+y) P(x, y)) and vx 3y P(x, y) (c) 3x³ (P(x, y) AP(y, x)) and 3x3 P(x, y)
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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Transcribed Image Text:Show that the two quantified statements in each problem are not logically equivalent by filling in a table so that, for the domain (a, b, c),
the values of the predicate P you select for the table causes one of the statements to be true and the other to be false. For example,
the table below shows that vx vy P(x, y) and 3x 3y P(x, y) are not logically equivalent because for the given values of the predicate P, vx
vy P(x, y) is false and 3x 3y P(x, y) is true.
P
a
EXERCISE 1.10.3: Showing non-equivalence for expressions with nested quantifiers.
b
a b C
T T T
T F T
C
T T F
(a) vx ³y P(x, y) and 3x vy P(x, y)
(b) vx³y ((x #y) A P(x, y)) and vx 3 P(x, y)
(c) 3x ³у (P(x, y) A P(y, x)) and 3x3y P(x, y)
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