# disjunctive/conjunctive form and Truths.

#### Sticksz

##### New member
Just to start I am not 100% sure if this is the right thread but I could not find something for this type of math.
Secondly I have been stumped on this for days now and I only have one more attempt at this homework assignment and I would like to get these two correct but I can not for the life on me find this in the text book or online somewhere on how I do these two problems I have searched everywhere and cant figure out how to do these two.
lastly I linked each of the two problems in imgur so its a bit easier to read.

First question is: For this one I tried using different properties in the text book but i have not gotten it right a single time so maybe i'm just not doing it right i'm not sure.

Second question is: For this one I tried making a truth table twice and maybe I did not construct it correctly either time?

Thank you for any help you can offer I feel defeated with these two problems and I cant seem to find help for this in my textbook or oneline.

#### Subhotosh Khan

##### Super Moderator
Staff member
Just to start I am not 100% sure if this is the right thread but I could not find something for this type of math.
Secondly I have been stumped on this for days now and I only have one more attempt at this homework assignment and I would like to get these two correct but I can not for the life on me find this in the text book or online somewhere on how I do these two problems I have searched everywhere and cant figure out how to do these two.
lastly I linked each of the two problems in imgur so its a bit easier to read.

First question is: For this one I tried using different properties in the text book but i have not gotten it right a single time so maybe i'm just not doing it right i'm not sure.

Second question is: For this one I tried making a truth table twice and maybe I did not construct it correctly either time?

Thank you for any help you can offer I feel defeated with these two problems and I cant seem to find help for this in my textbook or oneline.
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

#### pka

##### Elite Member
Just to start I am not 100% sure if this is the right thread but I could not find something for this type of math.
lastly I linked each of the two problems in imgur so its a bit easier to read. First question is: For this one I tried using different properties in the text book but i have not gotten it right a single time so maybe i'm just not doing it right i'm not sure. Second question is: For this one I tried making a truth table twice and maybe I did not construct it correctly either time?
Even though I have taught logic courses thru graduate level symbolic logic, I find these questions to be confusing.
For the first I can tell you that [imath](p \Rightarrow \neg q) \equiv (\neg p \vee \neg q)[/imath]
I am less sure that I know what the writer of the second wants: [imath](p \wedge \neg p) \vee (q \vee F) \vee (T \wedge q) \vee (\neg q \wedge T)[/imath]
That is a disjunctive form the first factor of which, [imath]p\wedge\neg p[/imath], is always false so can be ignored.
But the second and third terms are true if [imath]q\equiv T[/imath], the last term is true if [imath]q\equiv F[/imath].
What does that tell us about a disjunction?

#### Dr.Peterson

##### Elite Member
First question is:
This is a direct application of a theorem (or maybe even a definition) you should have been given. Were you taught that [imath](p \Rightarrow q) \equiv (\neg p \vee q)[/imath]? If so, then you are just replacing [imath]q[/imath] with [imath]\neg q[/imath].

For this one I tried using different properties in the text book but i have not gotten it right a single time so maybe i'm just not doing it right i'm not sure.

Second question is:
For this one I tried making a truth table twice and maybe I did not construct it correctly either time?
Please show us the truth table you made, so we can see what is going wrong!

#### JeffM

##### Elite Member
If Tom is a cat, then Tom is not a dog.

That is always true because dogs and cats are different categories. Thus, any equivalent statement must always be true.

Tom is not a cat or Tom is a dog. Is that statement always true? No. Tom may be a human.

Tom is a cat and Tom is not a dog. Is that statement always true? No. Tom may be human.

Tom is not a cat and Tom is not a dog. Is that statement always true? No, it is false if Tom is a cat or if Tom is a dog.

Tom is a cat or Tom is a dog. Is that statement always true? No, Tom may be human.

Tom is not a cat or Tom is not a dog. Is that statement always true? Yes. There are three possibilities. If Tom is a cat, then it is true that Tom is not a dog. If Tom is a dog, then it is true that Tom is not a cat.If Tom is neither a cat nor a dog, then obviously the statement is true.

Undoubtedly, Dr. Peterson and pka are correct: truth tables and axioms and theorems let you deal with complex concatenations, but ultimately symbolic logic can always be confirmed by the right example.