B. by using Rules of Replacement 1. [p V (~ p ^q)] → (p A q) 2. {~ (pvq) v [(~ p^ q) v ~q]} +→ ~ (pvq)
B. by using Rules of Replacement 1. [p V (~ p ^q)] → (p A q) 2. {~ (pvq) v [(~ p^ q) v ~q]} +→ ~ (pvq)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
Section: Chapter Questions
Problem 1RP: Although Pythagoras preceded William Shakespeare by 2,000 years, the philosophy of the Pythagoreans...
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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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1fcb6a7-c3ff-421e-930e-1530935f1de0%2F58e48f0c-eead-4981-b7d1-e798f36d9519%2Fshtoz0q_processed.png&w=3840&q=75)
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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1fcb6a7-c3ff-421e-930e-1530935f1de0%2F58e48f0c-eead-4981-b7d1-e798f36d9519%2Fk40zqfl_processed.png&w=3840&q=75)
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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