1. Using Laws of the Predicate Calculus 0-26, prove the following: (27) [X⇒ XVY] V" 2. Using Laws of the Predicate Calculus 0-27, prove the following: (28) [X^Y ⇒ X] "^ ⇒" 3. Using Laws of the Predicate Calculus 0-28, prove the following: [XAY=XVY)
1. Using Laws of the Predicate Calculus 0-26, prove the following: (27) [X⇒ XVY] V" 2. Using Laws of the Predicate Calculus 0-27, prove the following: (28) [X^Y ⇒ X] "^ ⇒" 3. Using Laws of the Predicate Calculus 0-28, prove the following: [XAY=XVY)
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Hello, I'm trying to complete the following Discrete Maths questions.
- I've attached a screenshot of the questions.
- I've also attached a picture of the Laws Of The Predicate Calculus sheet we use.
- The number beside the question e.g. (27) represents the Law on the Predicate Calculus Sheet.
PLEASE NOTE: The question title informs you up to which law on the sheet you can use. e.g. Using Laws of the Predicate Calculus 0–26, prove the following: You CANNOT use laws 27+ in this scenario.
![1. Using Laws of the Predicate Calculus 0-26, prove the following:
(27)
[X⇒ XVY]
V"
2. Using Laws of the Predicate Calculus 0-27, prove
(28)
[X^Y ⇒ X]
"^⇒"
the following:
3. Using Laws of the Predicate Calculus 0-28, prove the following:
[X^Y ⇒ XVY]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2775e8c-94c2-43ee-8379-e9f8aff0b154%2F78632191-5473-4ebd-a13d-df53ce18d062%2Fznyk7ie_processed.png&w=3840&q=75)
Transcribed Image Text:1. Using Laws of the Predicate Calculus 0-26, prove the following:
(27)
[X⇒ XVY]
V"
2. Using Laws of the Predicate Calculus 0-27, prove
(28)
[X^Y ⇒ X]
"^⇒"
the following:
3. Using Laws of the Predicate Calculus 0-28, prove the following:
[X^Y ⇒ XVY]
![0
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= associative*
= symmetric*
= identity*
■ reflexive
true
v symmetric*
v associative*
v idempotent*
V/=*
V/EE
v/v
v zero
Golden Rule*
A symmetric
^ associative
A idempotent
A identity
absorption.0
absorption.1
V/A
A/V
A over=
A/==
strong MP
replacement
→ definition*
→ reflexive
→> true
➡V
A➡
shunting
⇒ to A=
⇒ over=
definition*
LAWS OF THE PREDICATE CALCULUS
false definition*
- over=*
- neg-identity
[(X=(Y=Z)) = ((X=Y)=Z)]
[X=Y=Y=X]
[X=true=X]
[X=X]
[true]
[Xv Y = YvX]
[Xv (YvZ) = (Xv Y) v Z]
[Xv X = X]
[Xv (Y=Z) = Xv Y = Xv Z]
[Xv (Y=Z=W) = Xv Y = Xv Z = Xv W]
[Xv (YvZ) = (XVY) v (Xv Z)]
[Xv true = true]
[X^ Y = X = Y = XvY]
[XAY = YAX]
[XA (YAZ) = (X^Y) ^ Z]
[X^X = X]
[X A true = X]
[X^ (XVY) = X]
[XV (X^Y) = X]
[XV (YAZ) = (XVY) ^ (X v Z)]
[XA (YV Z) = (X^ Y) V (X^Z)]
[XA (Y=Z) = X^ Y = X^Z = X]
[XA (Y=Z=W) = XAY = XAZ = XAW]
[X^ (X=Y) = X^Y]
[(X=Y) ^ (W=X) = (X=Y) ^ (W=Y)]
[X Y = Xv Y = Y]
[X→X]
[X→> true]
[X → XV Y]
[X^Y = X]
[XAY = Z = X…(Y=Z]
[X = Y = X^Y=X)
[X➡ (Y=Z) = XAY=X^Z]
[X+Y=X^Y = Y]
[X-Y = Y➡X]
[false=true]
[-(X=Y)=-X=Y]
[-X=X=false]
postulates are decorated with a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2775e8c-94c2-43ee-8379-e9f8aff0b154%2F78632191-5473-4ebd-a13d-df53ce18d062%2F8fheqp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:0
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8a
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= associative*
= symmetric*
= identity*
■ reflexive
true
v symmetric*
v associative*
v idempotent*
V/=*
V/EE
v/v
v zero
Golden Rule*
A symmetric
^ associative
A idempotent
A identity
absorption.0
absorption.1
V/A
A/V
A over=
A/==
strong MP
replacement
→ definition*
→ reflexive
→> true
➡V
A➡
shunting
⇒ to A=
⇒ over=
definition*
LAWS OF THE PREDICATE CALCULUS
false definition*
- over=*
- neg-identity
[(X=(Y=Z)) = ((X=Y)=Z)]
[X=Y=Y=X]
[X=true=X]
[X=X]
[true]
[Xv Y = YvX]
[Xv (YvZ) = (Xv Y) v Z]
[Xv X = X]
[Xv (Y=Z) = Xv Y = Xv Z]
[Xv (Y=Z=W) = Xv Y = Xv Z = Xv W]
[Xv (YvZ) = (XVY) v (Xv Z)]
[Xv true = true]
[X^ Y = X = Y = XvY]
[XAY = YAX]
[XA (YAZ) = (X^Y) ^ Z]
[X^X = X]
[X A true = X]
[X^ (XVY) = X]
[XV (X^Y) = X]
[XV (YAZ) = (XVY) ^ (X v Z)]
[XA (YV Z) = (X^ Y) V (X^Z)]
[XA (Y=Z) = X^ Y = X^Z = X]
[XA (Y=Z=W) = XAY = XAZ = XAW]
[X^ (X=Y) = X^Y]
[(X=Y) ^ (W=X) = (X=Y) ^ (W=Y)]
[X Y = Xv Y = Y]
[X→X]
[X→> true]
[X → XV Y]
[X^Y = X]
[XAY = Z = X…(Y=Z]
[X = Y = X^Y=X)
[X➡ (Y=Z) = XAY=X^Z]
[X+Y=X^Y = Y]
[X-Y = Y➡X]
[false=true]
[-(X=Y)=-X=Y]
[-X=X=false]
postulates are decorated with a
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