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)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 34EQ

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