Description
1 Propositional Logic 1
We ask a logician (who only tells the truth) about his sentimental life, and he answers the following two statements:
I love Ann or I love Beth.
If I love Ann, then I love Beth.
What can we conclude? Answer the following questions by “yes”, “no”, “unsure”.
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Does he love Ann?
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Does he love Beth?
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Does he love both?
Sample Answer:
no,no,no
2 Propositional Logic 2
Which of the following are correct?
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F alse j= T rue.
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T rue j= F alse.
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(A ^ B) j= (A , B).
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A , B j= A _ B.
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A , B j= :A _ B.
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(A ^ B) ) C j= (A ) C) _ (B ) C).
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(C _ (:A ^ :B)) ((A ) C) ^ (B ) C).
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(A _ B) ^ (:C _ 😀 _ E) j= (A _ B).
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(A _ B) ^ (:C _ 😀 _ E) j= (A _ B) ^ (:D _ E).
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(A _ B) ^ :(A ) B) is satis able.
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(A , B) ^ (:A _ B) is satis able.
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(A , B) , C has the same number of models as A , B for any xed set of proposition symbols that includes A; B; C.
VE 492 : Electronic #10 (Due July 31st, 2020 at 11:59pm)
Sample Answer:
a,b,c,d
3 Propositional Logic 3
We denote L0 the set of propositional logic sentences built from a set X of n propositional symbols.
we consider the following new formal languages, where some logical connectives are not allowed:
L1 is de ned as follows:
True and False are sentences of L1. All symbols of X are sentences of L1. If s; s0 are two sentences of L1, then :s, (s ^ s0 ), (s _ s0 ), and (s ) s0 ) are four sentences of L1.
L2 is de ned as follows:
True and False are sentences of L2. All symbols of X are sentences of L2. If s; s0 are two sentences of L2, then :s, (s ^ s0 ), and (s _ s0 ) are three sentences of L2.
L3 is de ned as follows:
True and False are sentences of L3. All symbols of X are sentences of L3. If s; s0 are two sentences of L3, then :s and (s ^ s0 ) are two sentences of L3.
L4 is de ned as follows:
True and False are sentences of L4. All symbols of X are sentences of L4. If s are two sentences of L4, then :s is a sentence of L4.
We consider the following binary relation between languages: L L’ i any sentences that can be expressed in L can equivalently be expressed in L’.
Answer “yes” or “no” the following questions.
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L1 L0
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L2 L0
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L3 L0
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L4 L0
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L0 L1
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L0 L2
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L0 L3
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L0 L4
Sample Answer:
no,no,no,no,no,no,no,no
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First-Order Logic 1
Are the following are valid (necessarily true) sentences?
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(9x x = x) ) (8y9z y = z).
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8x P (x) _ 😛 (x).
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8x Smart(x) _ (x = x).
Answer “Valid” or “Invalid” the following questions.
VE 492 : Electronic #10 (Due July 31st, 2020 at 11:59pm)
Sample Answer:
Valid,Valid,Valid
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First-Order Logic 2
This exercise uses the function M ap Color and predicates In(T; y), Borders(x; y), and Country(x), whose arguments are geographical regions, along with constant symbols for various regions. In each of the following we give an English sentence and a number of candidate logical expressions.
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Paris and Marseilles are both in France.
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In(P aris ^ M arseilles; F rance).
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In(P aris; F rance) ^ In(M arseilles; F rance).
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In(P aris; F rance) _ In(M arseilles; F rance).
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There is a country that borders both Iraq and Pakistan.
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9c Country(c) ^ Border(c; Iraq) ^ Border(c; P akistan).
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9c Country(c) ) [Border(c; Iraq) ^ Border(c; P akistan)].
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[9c Country(c)] ) [Border(c; Iraq) ^ Border(c; P akistan)].
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9c Border(Country(c); Iraq ^ P akistan).
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All countries that border Ecuador are in South America.
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8c Country(c) ^ Border(c; Ecuador) ) In(c; SouthAmerica).
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8c Country(c) ) [Border(c; Ecuador) ) In(c; SouthAmerica)].
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8c [Country(c) ) Border(c; Ecuador)] ) In(c; SouthAmerica).
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8c Country(c) ^ Border(c; Ecuador) ^ In(c; SouthAmerica).
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No region in South America borders any region in Europe.
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:[9c; d In(c; SouthAmerica) ^ In(d; Europe) ^ Borders(c; d)].
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8c; d [In(c; SouthAmerica) ^ In(d; Europe)] ) :Borders(c; d)].
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:8c In(c; SouthAmerica) ) 9d In(d; Europe) ^ :Borders(c; d).
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8c In(c; SouthAmerica) ) 8d In(d; Europe) ) :Borders(c; d).
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No two adjacent countries have the same map color.
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8x; y :Country(x) _ :Country(y) _ :Borders(x; y) _ :(M apColor(x) = M apColor(y)).
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8x; y (Country(x)^Country(y)^Borders(x; y)^:(x = y)) ) :(M apColor(x) = M apColor(y)).
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8x; y Country(x) ^ Country(y) ^ Borders(x; y) ^ :(M apColor(x) = M apColor(y)).
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8x; y (Country(x) ^ Country(y) ^ Borders(x; y)) ) M apColor(x 6= y).
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For each of the logical expressions, state whether it…
1 correctly expresses the English sentence;
2 is syntactically invalid and therefore meaningless;
3 is syntactically valid but does not express the meaning of the English sentence.
VE 492 : Electronic #10 (Due July 31st, 2020 at 11:59pm)
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