B. by using Rules of Replacement 1. [p V (~ p ^q)] → (p A q) 2. {~ (pvq) v [(~ p^ q) v ~q]} +→ ~ (pvq)

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Chapter2: Second-order Linear Odes
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Prove that the following are equivalent by using rules of replacement (picture attached)

 

RULES OF INFERENCE AND REPLACEMENT
I. Modus Ponens (M.P.)
2. Modus Tollens (M.T.)
3. Hypothetical Syllogism (H.S.)
P
.:ppr
4. Disjunctive Syllogism (D.S.)
5. Conjunction (Conj.)
6. Constructive Dilemma (CD.)
pvq
P
(P> 9) • (r> s)
pvr
.. qvs
7. Simplification (Simp.)
8. Absorption (Abs.)
9. Addition (Add.)
P
..p> (p• q)
*pvq
The following sets of logically equivalent expressions can replace each other wherever they occur:
10. DeMorgan's Theorems (De M.)
-(p • q) = ("pv -q)
(p v q) = (~p• -q)
(p v q) = (q v p)
(p• q) = (9 • P)
[p v (q v r]] = [[p v q) v r]
[p• (9 • r]] = [lp • q) • r]
[p• (q v r]] = [[p •q) v (p• r]]
[p v (q • r]] = [[p v q) • (p v r]]
II. Commutation (Com.)
12. Association (Assoc.)
13. Distribution (Dist.)
14. Double Negation (D.N.)
15. Transposition (Trans.)
16. Material Implication (Impl.)
17. Material Equivalence (Equiv.)
(p> q) = ("p v q)
(p= q) = [[p>q) • 9 Pl]
(p= q) = [[p• q) v (-p•-q]]
[lp• q) >] = [p> (q=r]
p= (p v p)
p= (p•P)
18. Exportation (Exp.)
19. Tautology (Taut.)
Transcribed Image Text:RULES OF INFERENCE AND REPLACEMENT I. Modus Ponens (M.P.) 2. Modus Tollens (M.T.) 3. Hypothetical Syllogism (H.S.) P .:ppr 4. Disjunctive Syllogism (D.S.) 5. Conjunction (Conj.) 6. Constructive Dilemma (CD.) pvq P (P> 9) • (r> s) pvr .. qvs 7. Simplification (Simp.) 8. Absorption (Abs.) 9. Addition (Add.) P ..p> (p• q) *pvq The following sets of logically equivalent expressions can replace each other wherever they occur: 10. DeMorgan's Theorems (De M.) -(p • q) = ("pv -q) (p v q) = (~p• -q) (p v q) = (q v p) (p• q) = (9 • P) [p v (q v r]] = [[p v q) v r] [p• (9 • r]] = [lp • q) • r] [p• (q v r]] = [[p •q) v (p• r]] [p v (q • r]] = [[p v q) • (p v r]] II. Commutation (Com.) 12. Association (Assoc.) 13. Distribution (Dist.) 14. Double Negation (D.N.) 15. Transposition (Trans.) 16. Material Implication (Impl.) 17. Material Equivalence (Equiv.) (p> q) = ("p v q) (p= q) = [[p>q) • 9 Pl] (p= q) = [[p• q) v (-p•-q]] [lp• q) >] = [p> (q=r] p= (p v p) p= (p•P) 18. Exportation (Exp.) 19. Tautology (Taut.)
Part 2. Prove that the following are equivalent
A. by Constructing a truth table
1. [p V (~p ^q)] + (p ^ q)
2. {~ (pvq) v [(~ p^ q) v ~q]} «→ ~ (pvq)
B. by using Rules of Replacement
1. [p V (~ p ^q)] → (p ^ q)
2. {~ (pvq) v [(~ p^ q) v ~q]} «→ ~ (pvq)
Transcribed Image Text:Part 2. Prove that the following are equivalent A. by Constructing a truth table 1. [p V (~p ^q)] + (p ^ q) 2. {~ (pvq) v [(~ p^ q) v ~q]} «→ ~ (pvq) B. by using Rules of Replacement 1. [p V (~ p ^q)] → (p ^ q) 2. {~ (pvq) v [(~ p^ q) v ~q]} «→ ~ (pvq)
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