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Харьковский национальный университет радиоэлектроники

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Дискретная математика

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Discrete math with computers_3

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C.6. PROPOSITIONAL LOGIC

401

•Since P Q and P R are WFFs, (P Q) (P R) is a WFF.

•Since (P Q) (P R) and P (Q R) are WFFs, (P Q) (P R) ↔ P (Q R) is a WFF.

P	Q	R	P Q	P R			Q R	(P Q) (P R)
True	True	True	True	True			True	True
True	True	False	True	True			True	True
True	False	True	True	True			True	True
True	False	False	True	True			False	True
False	True	True	True	True			True	True
False	False	True	False	True			True	False
False	False	False	False	False			False	False

P	Q	R	P (Q R)		(P Q) (P R) ↔ P (Q R)
True	True	True	True			True
True	True	False	True			True
True	False	True	True			True
True	False	False	False			False
False	True	True	False			False
False	False	True	False			True
False	False	False	False			True

The proposition is satisﬁable but not a tautology.

•P and Q are WFFs.

•Since P and Q are WFFs, ¬P and ¬Q are WFFs.

• Since P , Q, ¬P and ¬Q are WFFs, P ¬Q and Q ¬P are WFFs.

•Since P and Q are WFFs, P ↔ Q is a WFF.

•Since P ↔ Q is a WFF, ¬(P ↔ Q) is a WFF.

•Since P ¬Q and Q ¬P are WFFs, (P ¬Q) (Q ¬P ) is a WFF.

• Since (P ¬Q) (Q ¬P ) and ¬(P ↔ Q) are WFFs, ((P ¬Q) (Q ¬P )) → ¬(P ↔ Q) is a WFF.

P	Q	¬P	¬Q	P ¬Q	Q ¬P
True	True	False	False	False	False
True	False	False	True	True	False
False	True	True	False	False	True
False	False	True	True	False	False

402			APPENDIX C. SOLUTIONS TO SELECTED EXERCISES

	P	Q		(P ¬Q) (Q ¬P )	P ↔ Q	¬(P ↔ Q)
	True	True		False	True	False
	True	False		True	False	True
	False	True		True	False	True
	False	False		False	True	False


	P	Q		((P ¬Q) (Q ¬P )) → ¬(P ↔ Q)
	True	True		True
	True	False		True
	False	True		True
	False	False		True

The proposition is a tautology.

7.(P → Q) (P → ¬Q)

•P and Q are WFFs.

•Since Q is a WFF, ¬Q is a WFF.

•Since P , Q, and ¬Q are WFFs, P → Q and P → ¬Q are WFFs.

•Since P → Q and P → ¬Q are WFFs, (P → Q) (P → ¬Q) is a WFF.

P	Q	¬Q	P → Q	P → ¬Q	(P → Q) (P → ¬Q)
True	True	False	True	False	False
True	False	True	False	True	False
False	True	False	True	True	True
False	False	True	True	True	True

The proposition is satisﬁable but not a tautology.

8.(P → Q) (¬P → Q)

•P and Q are WFFs.

•Since P is a WFF, ¬P is a WFF.

•Since P , Q, and ¬P are WFFs, P → Q and ¬P → Q are WFFs.

•Since P → Q and ¬P → Q are WFFs, (P → Q) (¬P → Q) is a WFF.

P	Q	¬P	P → Q	(¬P ) → Q	(P → Q) (¬P → Q)
True	True	False	True	True	True
True	False	False	False	True	False
False	True	True	True	True	True
False	False	True	True	False	False

The proposition is satisﬁable but not a tautology.

C.6. PROPOSITIONAL LOGIC

403

9.(P → Q) ↔ (¬Q → ¬P )

•P and Q are WFFs.

•Since P and Q are WFFs, ¬P and ¬Q are WFFs.

•Since P , Q, ¬Q, and ¬P are WFFs, P → Q and ¬Q → ¬P are WFFs.

•Since P → Q and ¬Q → ¬P are WFFs, (P → Q) ↔ (¬Q → ¬P ) is a WFF.

P	Q	¬P	¬Q	P → Q	¬Q → ¬P	(P → Q) ↔ (¬Q → ¬P )
True	True	False	False	True	True	True
True	False	False	True	False	False	True
False	True	True	False	True	True	True
False	False	True	True	True	True	True

The proposition is a tautology.

10.

{ I}

RQ R

{ I}

P(Q R)

11.The proof of y will have a symmetrical shape, but the proof of x will appear triangular, with more inference on the right side than on the left. In the general case, the proof of x with 2n variables will have height 2n, because every extra variable will require one extra inference above everything else. In contrast, the proof of y with 2n variables will have height n.

12.

(P Q) R			(P Q) R		{ EL}
			P Q
		{ EL}		{ ER}		(P Q) R	{ ER}
P Q
	{ EL}		Q			R		{ I}
					Q R
P

{ I}

P (Q R)

13.

P P → Q

{→E}

		{ I}
P Q		{ I}
P Q		P Q → R S		E
			{→		}
	R S

{ ER}

404	APPENDIX C. SOLUTIONS TO SELECTED EXERCISES

15.

{ I}

QP Q

{→I}

Q → P Q

18.

{ I}

P Q

{ IL}

(P Q) (Q R)

19. We prove that True True → True and then that True → True True, without translating True into False → False.

True True

{ ER}

True

{→I}

True True → True

True True

{ I}

True True

{→I}

True → True True

20. We prove that True False → True and then that True → False True.

	True		False
		{ID}		{CTR}
True False True			True			E
					{		}

True

{→I}

True False → True

True

{ IL}

True False

{→I}

True → True False

23.

•P is represented by P

•Q FALSE is represented by Or Q FALSE

C.6. PROPOSITIONAL LOGIC

405

• Q → (P → (P Q)) is represented by Imp Q (Imp P (And P Q))

25. Proof by equational reasonong.

(P False) (Q True)	{ null}
= False (Q True)	{ null}
= False Q	{ identity}
= Q False	{commutativity}
= Q	{ identity}
27. Proof by equational reasoning.
(P ((Q R) Q)) S	{ commutative}
= S (P ((Q R) Q))	{ commutative}
= S (((Q R) Q) P )	{ commutative}
= S ((Q (R Q)) P )	{ associative}
= S ((Q (Q R)) P )	{ commutative}
= S (((Q Q) R) P )	{ associative}
= S ((Q R) P )	{ idempotent}
= S ((R Q) P )	{ commutative}
30. Proof by equational reasoning.
(A B) B
= { identity}
(A B) (B False)
= { comm}
(B A) (B False)
= { over }
B (A False)
= { null }
B False
= { identity}
B

31. Solution by equational reasoning.

406 APPENDIX C. SOLUTIONS TO SELECTED EXERCISES

(¬A B) (A ¬B) = { over }

((¬A B) A) ((¬A B) ¬B) = { comm}

(A (¬A B)) (¬B (¬A B)) = { over }

((A ¬A) (A B)) ((¬B ¬A) (¬B B)) = { comm}

((A ¬A) (A B)) ((¬B ¬A) (B ¬B)) = { compl }

(True (A B)) ((¬B ¬A) True) = { comm}

((A B) True) ((¬B ¬A) True)

={ identityappliedtwice}

(A B) (¬B ¬A)

={DM }

(A B) ¬(B A) = { comm}

(A B) ¬(A B)

32. The problem is solved by equational reasoning:

¬(A B)

={double negationappliedtwice} ¬(¬¬A ¬¬B)

={DM }

¬(¬(¬A ¬B))

= {double negation} ¬A ¬B

33. Solution by equational reasoning.

(A B) (¬A C) (B C) = { identity}

(A B) (¬A C) ((B C) False) = { compl }

(A B) (¬A C) ((B C) (A ¬A))

= { over }

(A B) (¬A C) ((B C A) (B C ¬A)) = { comm}

(A B) (¬A C) ((A B C) (¬A B C))

= { comm}

(¬A C) (A B) ((A B C) (¬A B C)) = { assoc}

(¬A C) ((A B) (A B C)) (¬A B C)

C.6. PROPOSITIONAL LOGIC

407

= { comm}

((A B) (A B C)) (¬A C) (¬A B C) = { assoc}

((A B) (A B C)) ((¬A C) (¬A B C)) = { over }

((A (B (B C))) (¬A (C (B C)))

= { comm}

(A ((B C) B)) (¬A ((B C) C)) = { comm}

(A ((C B) B)) (¬A ((B C) C)) = {absorbtion, appliedtwice}

(A B) (¬A C)

34. A solution in natural deduction style:

		A ¬A	{ EL}			A ¬A		{ ER}
			{ EL}					{ ER}
		A					¬A			E
									{→		}
				False
		A solution in the proof-checker notation:
hwThm1 = Theorem [A ‘And‘ (Not A)] (FALSE)
proof_hwThm1 =
	(Assume(A ‘And‘ (Not A))
{----------------------------										} ‘AndEL‘
				A,
							Assume(A ‘And‘ (Not A))
{---------------------------------											} ‘AndER‘
							(Not A))
{------------------------------------------------------											}‘ImpE‘
							(FALSE)
35.
hwThm2 = Theorem [A] ((A ‘Imp‘ FALSE) ‘Imp‘ FALSE)
proof_hwThm2 =
		(Assume A,
							Assume(A ‘Imp‘ FALSE))
{---------------------------------------------------											} ‘ImpE‘
							FALSE
{-------------------------------------------------------											} ‘ImpI‘
		((A ‘Imp‘ FALSE) ‘Imp‘ FALSE)
	A A → False				E
				{→			}

False

{→I}

(A → False) → False

408	APPENDIX C. SOLUTIONS TO SELECTED EXERCISES

36.

hwThm3 = Theorem [A, A ‘Imp‘ B, B ‘Imp‘ C, C ‘Imp‘ D] (D)

proof_hwThm3 =

(((Assume A, Assume(A ‘Imp‘ B))

{--------------------------	} ‘ImpE‘
B,	Assume(B ‘Imp‘ C))
{--------------------------------------	} ‘ImpE‘
C,	Assume(C ‘Imp‘ D))
{----------------------------------------------	} ‘ImpE‘
	D

A A → B

{→E}

BB → C

		{→E}
		{→E}
C		C → D		E
			{→		}
	D

37.

hwThm6 = Theorem [A ‘Imp‘ B] ((B ‘Imp‘ FALSE) ‘Imp‘ (A ‘Imp‘ FALSE))

proofThm6 =

(((Assume A, Assume(A ‘Imp‘ B))

{-----------------------------	} ‘ImpE‘
B),	Assume(B ‘Imp‘ FALSE))
{-------------------------------------------	} ‘ImpE‘
	FALSE
{-----------------------------------	} ‘ImpI‘
(A ‘Imp‘ FALSE)
{-----------------------------------	} ‘ImpI‘

((B ‘Imp‘ FALSE) ‘Imp‘ (A ‘Imp‘ FALSE))

A A → B

{→E}

B → False

{→E}

False

{→I}

A → False

{→

}

(B → False) → (A → False)

38.

A A → ¬B

A A → B

{→

}

¬B

{→

}

{→

}

False

{→I}

A → False

C.6. PROPOSITIONAL LOGIC

409

39.

A A → False

{→

}

{→

}

False

{CTR}

False

{CTR}

¬B

A → B

{→

}

A → ¬B

{→

}

{

}

41.

(A → B) (A → ¬B)

P Q

R S

{ I}

P Q

R S

(P Q) (R S)

{

}

42.

P Q

{ EL}

P → R

{→

}

{→I}

PQ → R

43.We prove that True True → True and then that True → True True.

	True		True
		{ID}		{ID}
		{ID}		{ID}
True True True			True			E
	True				{		}
	True

{→I}

True True → True

True

{ IL}

True True

{→I}

True → True True

45. A list comprehension can generate a truth table for you.

logicExprValue1 = [((a,b),logicExpr1 a b) | a <- [False,True],

b <- [False,True]

]

logicExprValue2 = [((a,b),logicExpr2 a b) | a <- [False,True],

b <- [False,True]

]

410	APPENDIX C. SOLUTIONS TO SELECTED EXERCISES

46.

logicExprValue3 = [((a,b,c),logicExpr3 a b c) | a <- [False,True],

b <- [False,True], c <- [False,True]

]

logicExprValue4 = [((a,b,c),logicExpr4 a b c) | a <- [False,True],

b <- [False,True], c <- [False,True]

]

47.

distribute :: Logic -> Logic

distribute (And a (Or b c)) = Or (And a b) (And a c) distribute (Or a (And b c)) = And (Or a b) (Or a c)

deMorgan :: Logic -> Logic

deMorgan (Not (Or a b)) = And (Not a) (Not b) deMorgan (Not (And a b)) = Or (Not a) (Not b) deMorgan (And (Not a) (Not b)) = Not (Or a b) deMorgan (Or (Not a) (Not b)) = Not (And a b)

48.

(C A B) C	{ commutative}
= C (C A B)	{ commutative}
= (C C) (C A) (C B)	{ distributes over }
= C (C A) (C B)	{ idempotent}
= C ((C A) (C B))	{ associative}
= C (C (A B))	{ distributes over }

49.

C (A (B C))	{ distributes over }
= C ((A B) (A C))	{ distributes over }
= C ((A C) (A B))	{ commutative}
= (C (A C)) (A B)	{ associative}
= ((C A) (C C)) (A B))	{ distributes over }
= ((C A) C) (A B)	{ idempotent}

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