Exercise 6.1 For each of the following wffs, give a countermodel for every system in which it is not valid, and give a semantic validity proof for every system in which it is valid. When you use a single countermodel or validity proof for multiple systems, indicate which systems it is good for. a)* □[P→◇(Q→R)]→◇[Q→(□P→◇R)] b) ◇(Р^◇Q)→(□◇Р→◇□Q) c) (PVOQ)-(□PV◇Q) d)* □(P↔Q)→□(□P↔˜¯Q) e) □(P^Q)→□□(◇P→◇Q) f) □(¯Р→Q)→□(□P→¯Q) g)* ◇◇□PP h) ◇◇P→□◇P i) □[□(P→□P)→¯P]→(◇□P→¯P)
Exercise 6.1 For each of the following wffs, give a countermodel for every system in which it is not valid, and give a semantic validity proof for every system in which it is valid. When you use a single countermodel or validity proof for multiple systems, indicate which systems it is good for. a)* □[P→◇(Q→R)]→◇[Q→(□P→◇R)] b) ◇(Р^◇Q)→(□◇Р→◇□Q) c) (PVOQ)-(□PV◇Q) d)* □(P↔Q)→□(□P↔˜¯Q) e) □(P^Q)→□□(◇P→◇Q) f) □(¯Р→Q)→□(□P→¯Q) g)* ◇◇□PP h) ◇◇P→□◇P i) □[□(P→□P)→¯P]→(◇□P→¯P)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.3: De Moivre’s Theorem And Roots Of Complex Numbers
Problem 28E
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I ONLY NEED 6.1b and 6.1h thanks
![Exercise 6.1 For each of the following wffs, give a countermodel
for every system in which it is not valid, and give a semantic validity
proof for every system in which it is valid. When you use a single
countermodel or validity proof for multiple systems, indicate which
systems it is good for.
a)* □[P→◇(Q→R)]→◇[Q→(□P→◇R)]
b) ◇(Р^◇Q)→(□◇Р→◇□Q)
c) (PVOQ)-(□PV◇Q)
d)* □(P↔Q)→□(□P↔˜¯Q)
e) □(P^Q)→□□(◇P→◇Q)
f) □(¯Р→Q)→□(□P→¯Q)
g)* ◇◇□PP
h) ◇◇P→□◇P
i) □[□(P→□P)→¯P]→(◇□P→¯P)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2Fc9167d1d-2bda-4cbd-a657-8947359f7fb8%2Ffbfyzka_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 6.1 For each of the following wffs, give a countermodel
for every system in which it is not valid, and give a semantic validity
proof for every system in which it is valid. When you use a single
countermodel or validity proof for multiple systems, indicate which
systems it is good for.
a)* □[P→◇(Q→R)]→◇[Q→(□P→◇R)]
b) ◇(Р^◇Q)→(□◇Р→◇□Q)
c) (PVOQ)-(□PV◇Q)
d)* □(P↔Q)→□(□P↔˜¯Q)
e) □(P^Q)→□□(◇P→◇Q)
f) □(¯Р→Q)→□(□P→¯Q)
g)* ◇◇□PP
h) ◇◇P→□◇P
i) □[□(P→□P)→¯P]→(◇□P→¯P)
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