EQUIVALENCE LAWS
- using the equivalence laws, we’ll
show that the sentences on the left hand side and right hand side of the “↔” symbol
are equivalent.
Examples:
1. [ ~(~P) ^ (Q v P) ] ↔ P v P
Statements Reasons
[ ~(~P) ^ (Q v P) ] ↔ P v P Given
[ ~(~P) ^ (P v Q) ] ↔ P v P Commutativity
[ P ^ (P v Q) ] ↔ P v P Double Negation
P ↔ P v P Absorption
P v P
↔ P v P Idempotency
Equal
2. ~ { ~ [ P ^ (P v Q) ] } v ~ (P v
P) ↔ T
Statements Reasons
~ { ~ [ P ^ (P v Q) ] } v ~ (P v P) ↔ T Given
~ (~P ) v ~ (P v P) ↔ T Absorption
P
v ~ (P v P) ↔ T Double Negation
P
v ~P ↔ T Idempotency
T
↔ T Inverse
Equal
3. (~Q → ~P) ^ [ ~Q v {~ (~ (P v (P
^ Q)))} ] ↔ (P ↔ Q)
Statements Reasons
(~Q → ~P) ^ [ ~Q v {~ (~ (P v (P ^ Q)))} ] ↔ (P ↔ Q) Given
(P → Q) ^ [ ~Q v {~ (~ (P v (P ^ Q)))}
] ↔ (P ↔ Q) Contrapositive Law
(P → Q) ^ [ ~Q v { (P v (P ^ Q)
} ] ↔ (P ↔ Q) Double Negation
(P
→ Q) ^ [ ~Q v (P) ] ↔ (P ↔ Q)` Absorption
(P → Q) ^ (Q
→ P) ↔ (P ↔ Q)` Material Implication
(P
↔ Q)` ↔ (P ↔ Q)` Material Equivalence
Equal
Method of Proof in Proposition
Logic
Chain Reasoning
-
A chain reasoning is also called as a “direct proof”. You have to use one or
more of Inference Rules and/or the Equivalence Laws in proving.
Examples:
a.) Q v ~P
~R v ~Q (Q v ~P) ^ (~R v ~Q) → (P → ~R)
\ P → ~R
Statements Reasons
1.) ~R v ~Q Premise
2.) R → ~Q Material
Implication
3.) Q → ~R Contrapositive
Law
4.) Q v ~P Premise
5.) ~Q → ~P Material
Implication
6.) P → Q Contrapositive
Law
7.) (P → Q) ^ (Q → ~R) Conjunction (6,3)
8.) P → ~R Hypothetical
Syllogism
b.) ~P v Q
S v ~R
~ (Q ^ S) (~P v Q) ^ (S v ~R) ^ ~ (Q v S) → ~ ( P ^ R)
\ ~ (P ^ R)
Statements Reasons
1.) ~ (Q ^ S) Premise
2.) ~Q v ~S De
Morgan’s Law
3.) Q → ~S Material
Implication
4.) S → ~Q Contrapositive
Law
5.) S v ~R Premise
6.) ~R v S Commutativity
7.) R → S Material
Implication
8.) (R → S) ^ (S → ~Q) Conjunction (7,4)
9.) R → ~Q Hypothetical
Syllogism
10.) ~P v Q Premise
11.) P → Q Material
Implication
12.) ~Q → ~P Contrapositive
Law
13.) (R → ~Q) ^ (~Q → ~P) Conjunction (9,12)
14.) R → ~ P Hypothetical
Syllogism
15.) ~R v ~P Material
Implication
16.) ~P v ~R Commutativity
17.) ~ (P ^ R) De
Morgan’s Law
c.) ~A v ~B
A
~J
J → Y
(~A v ~B) ^ A ^ ~J ^ (J → Y) → ~ (~J → B)
\ ~ (~J → B)
Statements Reasons
1.) J → Y Premise
2.) ~J v Y Material
Implication
3.) ~J Premise
4.)
(~J v Y) ^ ~J Conjunction
(2,3)
5.)
~J Absorption
6.)
~A v ~B Premise
7.)
A → ~B Material
Implication
8.) A Premise
9.)
(A → ~B) ^ A Conjunction
(7,8)
10.)
~B Modus
Ponens
11.)
~J ^ ~B Conjunction
(5,10)
12.)
~ (J v B) De
Morgan’s Law
13.) ~ (~J → B) Material
Implicatio
good thanks for this
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