Proposition Logic doubt

Question

Given: (p$ \vee$ q) is True. 

Find the truth value of statements, 

1. p is false or q is true. (Can't determine) 

2. If p is false then q is true. (True) 

is my answer correct????? 

Comments

@ankitgupta.1729

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I guess you are finding the truth values of given statements. So, truth value can be True or False (But not both).

If we have to find the truth value of a statement which is written in the language of first order logic then we need the domain but it is not cleared why condition “$(p \vee q)$ is true” is given. It might be happened that domain of p is dependent on domain of q and in other way, so we might take domain for p,q as

(p=False,q=True), (p=True,q=False) and (p=True,q=True).  

$1.$ p is false and q is true

Taking $p$ and $q$ as propositional variables and $p$ as False and $q$ as True.

Say, $F(p)$ means $p$ is false and $T(q)$ means $q$ is true.

Since “and” is sentential connective, so you can write the given statement as:

$F(p) \wedge T(q)$

So, if $p$ is False and $q$ is True then given statement is true because then $F(p)$ means “False is False” which is equivalent to expression “a is b” which means $a=b$ and so given statement becomes true.

Same you can do for other elements from the domain and do the same for other statement.

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@ankitgupta.1729  i have edited question, please check again and solve using only proposition logic. 

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what are $p$ and $q$ ?

Your statement is written in a natural language and it is a compound statement of two statements “p is false” and “q is true”. Now to determine the truth value of both statements, we have to know whether $p$ is variable or constant or some name or it belongs to the set of mathematical expressions like 1+1=2 (These are called “Terms” in logic)

Without knowing what p and q are, how can we determine the truth values of these statements.

Now, it is given that $p \vee q$ is true then what are the propositions $p$ and $q$ ?

In propositional logic, we convert a statement which is written in natural language into a mathematical language with connectives,parenthesis etc. So, we have to know what p and q are.

Usually, we use letters x,y,z,p,q,… for variables which I have used in the above comment.

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Question is like, A result of a exam is out and the result is Jack or Jill pass?

p: Jack is pass. q:Jill is pass.

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@ankitgupta.1729 please reply..... 

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“The result of an exam is out and the result is Jack has passed the exam or the result is Jill has passed the exam”

R: The result of an exam is out

P:  the result is Jack has passed the exam

Q: the result is Jill has passed the exam

$R \wedge (P \vee Q)$ or $(R \wedge P) \vee Q$

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No, given statement is "Jack has passed the exam or Jill has passed the exam".

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Then it is $P \vee Q$

where P:  Jack has passed the exam and Q:  Jill has passed the exam

“or” in natural language can be used for “inclusive or” or “exclusive or” and here it seems an “inclusive or” is intended.

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Yes. So Given information is (P $\vee$ Q) is True.

Now , check my answer for statement 1 & 2. is it correct??

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As you have given $P$ and $Q$ then we can say the truth value of your statement 1 exists but we don’t know what it is because we don’t know the truth values of $P$ and $Q.$

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(P $\vee$ Q)  is True that's mean atleast one is True.

or we can say,

P is True or Q is True or both are True.

on the basis of this information, can we determine truth value of given statements in question?

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No. As I already said that we don’t know the truth values of $P$ and $Q$ it means we can’t determine the truth value of your statement 1. You might have to define like “who is Jack/Jill” and “what exam it was” etc otherwise it is difficult to know the truth values of P and Q.

If P and Q are mathematical expressions like $1+1=2$(in decimal number system) then it is easy to determine the truth values but here it is difficult to know the truth values for P and Q.

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I already defined meaning of p and q in comment. 

https://gateoverflow.in/403660/proposition-logic-doubt?show=403784#c403784

p: Jack is pass.

q:Jill is pass

Answer 1

The statement "If (p ∧ q) is True, then if p is false then q is true" is true.

We know that (p ∧ q) is true, which means that both p and q are true. If p were false, then (p ∧ q) would be false, which contradicts our initial assumption. Therefore, we can conclude that p must be true.

Now, if p is true, then the conditional statement "if p is false then q is true" is vacuously true, since the antecedent (p is false) is false.

Therefore, the statement "If (p ∧ q) is True, then if p is false then q is true" is true.

The first statement, "p is false and q is true," cannot be determined from the fact that (p ∧ q) is true, since there are other possibilities for p and q to be true simultaneously.