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kenneth h rosen chapter 1 section 1.5 nested quantifers question 34
ykrishnay
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Apr 18, 2022
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Find a common domain for the variables x, y, and z
for which the statement ∀x∀y((x = y) → ∀z((z = x) ∨
(z = y))) is true and another domain for which it is false.
discrete-mathematics
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ykrishnay
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shishir__roy
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Apr 19, 2022
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Given statement is True when x, y, z $\in$ {1} and is False when x, y, z $\in$ {1, 2};
counter example (1=1) $\implies$ ( (2=1) $\lor$ (2=1) ) is False ie statement is False when x = y = 1 and z = 2.
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Arjun
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Apr 19, 2022
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Is the given statement true only for a singleton domain and false for every other domain?
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Yes, for every domain with more than 1 element the statement will be False.
And for every singleton domain the statement will be True.
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kenneth h rosen chapter 1 section nested quantifers excercise 1.5 question 40
Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) ∀x∃y(x = 1/y) b) ∀x∃y(y^2 − x < 100)
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kenneth h rosen chapter 1 section "Nested quantifers" excercise 1.5 question 26's g
Let Q(x, y) be the statement “x + y = x − y.” If the do- main for both variables consists of all integers, what are the truth values? g) ∃y∀xQ(x, y) Basically i done all the subquestions (a,b,c,d,e,f,h,i) from this question but confused in g subquestion please give answer
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kenneth h rosen chapter 1 section section 1.5 nested quatnifiers excercise 49
49. a) Show that ∀xP (x) ∧ ∃xQ(x) is logically equivalent to ∀x∃y (P (x) ∧ Q(y)), where all quantifiers have the same nonempty domain. b) Show that ∀xP (x) ∨ ∃xQ(x) is equivalent to ∀x∃y (P (x) ∨ Q(y)), where all quantifiers have the same nonempty domain. please anybody tell how to prove this logical equivalency ?
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kenneth h rosen chapter 1 section 1.5 nested quantifiers excercise 1.5 question 48
Show that ∀xP (x) ∨ ∀xQ(x) and ∀x∀y(P (x) ∨ Q(y)), where all quantifiers have the same nonempty domain, are logically equivalent. (The new variable y is used to combine the quantifications correctly.)
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