a. ~W -> ~J

c. ~H -> ~J

e. ~(J & M)

g. ~~O -> ~(L & W) (or I would accept O -> ~(L&W))

i. O -> ~(J v M) (or I would accept O -> (~J & ~M))

k. (~J -> M) & (~~J -> ~M) (again, I would accept J rather than ~~J)
 
 

Set 3.

a. (D v E) -> G

c. ~M -> ~(D&E)

e. (S v R) -> (~(W & ~D) -> ~G)

g. (G <-> ~(D v E)) ->((S v R) & ~W)

i. ~(M & ~T) -> ~(D & E)
 
 

Set 4.

a. ~(L v M) -> ~G

c. (~N -> ~R) & (~F -> ~N)

e. ~(G v C) -> S

g. O -> ~(S & G)

i. ~R -> ((L & ~C) -> (S & G))
 
 
 
 

Truth Table Exercises

I will assume that most people wanted to do full truth tables of the other set of problems I handed out, even though it suggested assuming some propositions were true and others false and then working from there. Here, the truth tables are enormously time-comsuming in HTML, so I will do every third one.

c. (A & X) v (B & Y)

(A	&	X)		v		(B	&	Y)
T	T	T		T		T	T	T
T	T	T		T		T	F	F
T	T	T		T		F	F	T
T	T	T		T		F	F	F
T	F	F		T		T	T	T
T	F	F		F		T	F	F
T	F	F		F		F	F	T
T	F	F		F		F	F	F
F	F	T		T		T	T	T
F	F	T		F		T	F	F
F	F	T		F		F	F	T
F	F	T		F		F	F	F
F	F	F		T		T	T	T
F	F	F		F		T	F	F
F	F	F		F		F	F	T
F	F	F		F		F	F	F

 

g. (A <-> B) <-> (Z <-> X)

(A	<->	B)		<->		(Z	<->	X)
T	T	T		T		T	T	T
T	T	T		F		T	F	F
T	T	T		F		F	F	T
T	T	T		F		F	F	F
T	F	F		T		T	F	T
T	F	F		T		T	F	F
T	F	F		T		F	F	T
T	F	F		F		F	T	F
F	F	T		F		T	T	T
F	F	T		T		T	F	F
F	F	T		T		F	F	T
F	F	T		F		F	T	F
F	T	F		F		T	F	T
F	T	F		F		T	F	F
F	T	F		F		F	F	T
F	T	F		T		F	T	F

 

j. (~X&~Y) <-> (A -> ~(X & Y))

(~

X

&

~

Y)

<->

(A

->

~

(X

&

Y)

F

T

F

F

T

T

T

F

F

T

T

T

F

T

F

F

T

F

F

T

F

T

T

T

F

T

F

T

F

F

T

T

T

T

F

F

F

T

F

T

F

F

F

T

T

T

F

F

T

F

F

F

T

F

T

T

T

F

F

T

T

F

F

F

T

F

F

T

T

F

F

T

T

F

T

T

F

T

T

T

T

F

F

F

T

F

T

T

F

T

F

T

T

F

F

F

 

m. ~(A -> (~A -> (~B v X)))

~	(A	->	(~	A	->	(~	B	v	X)))
F	T	T	F	T	T	F	T	T	T
F	T	T	F	T	T	F	T	F	F
F	T	T	F	T	T	T	F	T	T
F	T	T	F	T	T	T	F	T	F
F	F	T	T	F	T	F	T	T	T
F	F	T	T	F	F	F	T	F	F
F	F	T	T	F	T	T	F	T	T
F	F	T	T	F	T	T	F	T	F

 

p. ~~(A & ~(B -> ~(C -> ~(X v Y))))

~

~

(A

&

~

(B

->

~

(C

->

~

(X

v

Y))))

F

T

T

F

F

T

T

T

T

F

F

T

T

T

F

T

T

F

F

T

T

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F

F

T

T

F

F

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F

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F

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F

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F

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F

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F

F

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F

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T

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T

F

F

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T

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T

T

F

F

F

T

F

F

T

T

F

F

F

F

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F

F

F

T

T

T

T

F

F

T

T

T

F

T

F

F

F

T

T

T

T

F

F

T

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F

F

T

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F

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T

T

T

T

F

F

F

T

T

F

T

F

F

T

T

F

F

T

T

T

F

F

F

F

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F

F

F

T

T

T

F

T

F

T

T

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F

T

F

F

F

T

T

T

F

T

F

T

T

F

F

T

F

F

F

T

T

T

F

T

F

F

T

T

F

T

F

F

T

T

F

F

F

T

T

F

F

F

F

T

F

F

F

F

T

T

T

F

F

T

T

T

F

T

F

F

F

F

T

T

T

F

F

T

T

F

F

T

F

F

F

F

T

T

T

F

F

F

T

T

F

T

F

F

F

F

T

F

T

T

T

F

F

F

F

T

F

F

F

F

T

T

F

T

F

T

T

T

F

T

F

F

F

F

T

T

F

T

F

T

T

F

F

T

F

F

F

F

T

T

F

T

F

F

T

T

F

T

F

F

F

F

T

F

F

T

T

F

F

F

 

Let's move on to section 2.
 
 

a. (G v H) & ~A

(G	v	H)	&	~	A
T	T	T	F	F	T
T	T	T	T	T	F
T	T	F	F	F	T
T	T	F	T	T	F
F	T	T	F	F	T
F	T	T	T	T	F
F	F	F	F	F	T
F	F	F	F	T	F

d. (A v B) <->((G v H) -> A)

(A	v	B)	<->	((G	v	H)	->	A))
T	T	T	T	T	T	T	T	T
T	T	T	T	T	T	F	T	T
T	T	T	T	F	T	T	T	T
T	T	T	T	F	F	F	T	T
T	T	F	T	T	T	T	T	T
T	T	F	T	T	T	F	T	T
T	T	F	T	F	T	T	T	T
T	T	F	T	F	F	F	T	T
F	T	T	F	T	T	T	F	F
F	T	T	F	T	T	F	F	F
F	T	T	F	F	T	T	F	F
F	T	T	T	F	F	F	T	F
F	F	F	T	T	T	T	F	F
F	F	F	T	T	T	F	F	F
F	F	F	T	F	T	T	F	F
F	F	F	T	F	T	F	F	F

g. ~(X v ~Y) -> (G<->(~H & I))

~	(X	v	~	Y)	->	(G	<->	(~	H	&	I))
F	T	T	F	T	T	T	F	F	T	F	T
F	T	T	F	T	T	T	F	F	T	F	F
F	T	T	F	T	T	T	T	T	F	T	T
F	T	T	F	T	T	T	F	T	F	F	F
F	T	T	F	T	T	F	T	F	T	F	T
F	T	T	F	T	T	F	T	F	T	F	F
F	T	T	F	T	T	F	F	T	F	T	T
F	T	T	F	T	T	F	T	T	F	F	F
F	T	T	T	F	T	T	T	F	T	T	T
F	T	T	T	F	T	T	F	F	T	F	F
F	T	T	T	F	T	T	T	T	F	T	T
F	T	T	T	F	T	T	F	T	F	F	F
F	T	T	T	F	T	F	T	F	T	F	T
F	T	T	T	F	T	F	T	F	T	F	F
F	T	T	T	F	T	F	F	T	F	T	T
F	T	T	T	F	T	F	T	T	F	F	F
T	F	F	F	T	F	T	F	F	T	F	T
T	F	F	F	T	F	T	F	F	T	F	F
T	F	F	F	T	T	T	T	T	F	T	T
T	F	F	F	T	T	T	F	T	F	F	F
T	F	F	F	T	F	F	T	F	T	F	T
T	F	F	F	T	T	F	T	F	T	F	F
T	F	F	F	T	F	F	F	T	F	T	T
T	F	F	F	T	T	F	T	T	F	F	F
F	F	T	T	F	T	T	F	F	T	F	T
F	F	T	T	F	T	T	F	F	T	F	F
F	F	T	T	F	T	T	T	T	F	T	T
F	F	T	T	F	T	T	F	T	F	F	F
F	F	T	T	F	T	F	T	F	T	F	T
F	F	T	T	F	T	F	T	F	T	F	F
F	F	T	T	F	T	F	F	T	F	T	T
F	F	T	T	F	T	F	T	T	F	F	F

j.~((G v (H<-> ~A)) & ~A)

~	((G	v	(H	<->	~	A))	&	~	A)
T	T	T	T	F	F	T	F	F	T
F	T	T	T	T	T	F	T	T	F
T	T	T	F	T	F	T	F	F	T
F	T	T	F	F	T	F	T	T	F
T	F	F	T	F	F	T	F	F	T
F	F	T	T	T	T	F	T	T	F
T	F	T	F	T	F	T	F	F	T
F	F	F	F	F	T	F	F	T	F

 

SET 3.

a. (A & G) v (B & H)

(A	&	G)	v	(B	&	H)
T	T	T	T	T	T	T
T	T	T	T	T	F	F
T	T	T	T	F	F	T
T	T	T	T	F	F	F
T	F	F	T	T	T	T
T	F	F	F	T	F	F
T	F	F	F	F	F	T
T	F	F	F	F	F	F
F	F	T	T	T	T	T
F	F	T	F	T	F	F
F	F	T	F	F	F	T
F	F	T	F	F	F	F
F	F	F	T	T	T	T
F	F	F	F	T	F	F
F	F	F	F	F	F	T
F	F	F	F	F	F	F

d. (A v G) & (B v H)

(A	v	G)	v	(B	&	H)
T	T	T	T	T	T	T
T	T	T	T	T	F	F
T	T	T	T	F	F	T
T	T	T	T	F	F	F
T	T	F	T	T	T	T
T	T	F	T	T	F	F
T	T	F	T	F	F	T
T	T	F	T	F	F	F
F	T	T	T	T	T	T
F	T	T	T	T	F	F
F	T	T	T	F	F	T
F	T	T	T	F	F	F
F	F	F	T	T	T	T
F	F	F	F	T	F	F
F	F	F	F	F	F	T
F	F	F	F	F	F	F

g. (H <->G) -> (A <->B)

(H	<->	G)	->	(A	<->	B)
T	T	T	T	T	T	T
T	T	T	F	T	F	F
T	T	T	F	F	F	T
T	T	T	T	F	T	F
T	F	F	T	T	T	T
T	F	F	T	T	F	F
T	F	F	T	F	F	T
T	F	F	T	F	T	F
F	F	T	T	T	T	T
F	F	T	T	T	F	F
F	F	T	T	F	F	T
F	F	T	T	F	T	F
F	T	F	T	T	T	T
F	T	F	F	T	F	F
F	T	F	F	F	F	T
F	T	F	T	F	T	F

h. ~(X & G)

~	(X	&	G)
F	T	T	T
T	T	F	F
T	F	F	T
T	F	F	F