# Draw a complete truth-table for the following sentence

I need someone to tell me if I answered these questions correctly and if not explain what I did wrong. My answers are in bold.

1. Indicate which of the following are true and which are false in an interpretation in which A and B are false and C is true.
a) (A v (B ^ C)) (false)
b) (-A ^ (B v C)) (true)
c) ((A v B) ^ C) (false)
d) (-(A ^ B) ^ C) (true)
e) (C -> (A -> B)) (false)
f) (A -> (C -> B))) (false)
g) (-A -> (A <-> B)) (false)
h) ((A <-> B) <-> C) (true)
i) ((A <-> -B) <-> (B <-> -A)) (true)
j) (((A ^ C) -> B) <-> (B -> (A ^ C)) (false)

2. Draw a complete truth-table for the following sentence and indicate whether it is valid or invalid.
(((B ^ C) -> A) <-> ((B -> A) v (C -> A)))
I think it's Invalid

Last edited:

Related Introductory Physics Homework Help News on Phys.org
honestrosewater
Gold Member
robert said:
e) (C -> (A -> B)) (false)
f) (A -> (C -> B))) (false)
g) (-A -> (A <-> B)) (false)
j) (((A ^ C) -> B) <-> (B -> (A ^ C)) (false)
Try these again.
(T -> T) is T
(T -> F) is F
(F -> T) is T
(F -> F) is T
2. Draw a complete truth-table for the following sentence and indicate whether it is valid or invalid.
(((B ^ C) -> A) <-> ((B -> A) v (C -> A)))
I think it's Invalid
Under which interpretation?

honestrosewater said:
Try these again.
(T -> T) is T
(T -> F) is F
(F -> T) is T
(F -> F) is T
Under which interpretation?
Ya I wasn't sure what to do with if cases. I think I understand it now.

Doesn't valid mean it is true under all interpretations?

Here are my new answers now that I understand how if statements work.

1. Indicate which of the following are true and which are false in an interpretation in which A and B are false and C is true.
a) (A v (B ^ C)) (false)
b) (-A ^ (B v C)) (true)
c) ((A v B) ^ C) (false)
d) (-(A ^ B) ^ C) (true)
e) (C -> (A -> B)) (true)
f) (A -> (C -> B))) (true)
g) (-A -> (A <-> B)) (true)
h) ((A <-> B) <-> C) (true)
i) ((A <-> -B) <-> (B <-> -A)) (true)
j) (((A ^ C) -> B) <-> (B -> (A ^ C)) (true)

2. Draw a complete truth-table for the following sentence and indicate whether it is valid or invalid.
(((B ^ C) -> A) <-> ((B -> A) v (C -> A)))
I think it's valid

honestrosewater
Gold Member
Yes, unless I made mistakes, they're all correct now. But I suggest that you look up your definition of valid. The use of <-> in the statement gives me doubts. My definition of valid doesn't apply to statements but to arguments: an argument is valid iff there exists no interpretation where the premises are all true and the conclusion is false. IOW, if all of the premises are true, the conclusion must also be true (this allows for the case that all of the premises cannot be true together). So they should have used something to specify a set of premises and a conclusion. In using <->, I assume that they mean for ((B ^ C) -> A) and ((B -> A) v (C -> A)) to take turns as premise and conclusion, but you may want to make sure.