Question

If p is T and q is F then which of the following have the truth value T?
A. p $ \vee $ q
B. $ \sim $p$ \vee $q
C. p $ \vee $ ($ \sim $q)
D. p $ \wedge $ ($ \sim $q)

Answer 1

Hint: In the question, to get the correct answer, we have to proceed by going through the options. Using the relation given, we will solve the particular option and then after take the next option and check it into the corresponding mathematical expression and then check whether it True or False.

Complete step-by-step solution:
Given p′s truth value is T and q′s truth value is F.
We have to determine which of the given have the truth value T.
First of all, we will write the given relation:
Let us take the first option:
p $ \vee $ q
As we know, if p and q are statements, the disjunction of p and q (p $ \vee $ q) is false when both p and q are false, and is true otherwise.
As given in the question p is True and q is False.
Therefore, the disjunction of p and q (p $ \vee $ q) is True here.
Let us take Second option:
$ \sim $p $ \vee $ q
It's not true that,” so $ \sim $ P means “It's not true that P.” We often read $ \sim $ P simply as “not P.”
Since we already find that the disjunction of p and q (p $ \vee $ q) is True.
Therefore, $ \sim $p $ \vee $ q is False here.
Let us take Third option:
p $ \vee $ ($ \sim $q)
In this case p is True and $ \sim $q is also True.
And as we know, if p and q are statements, the disjunction of p and q (p $ \vee $ q) is false when both p and q are false, and is true otherwise.
Therefore, the disjunction of p and $ \sim $q (p $ \vee $ $ \sim $q) is True here
Let us take fourth option:
p $ \wedge $ ($ \sim $q)
As we know, if p and q are statements, the conjunction of p and q (p $ \wedge $ q) is true when both p and q are true, and is false otherwise.
In this case, p is True and $ \sim $ q is also true.
Therefore, the conjunction of p and q (p $ \wedge $ q) is True here.
Thus, From the above tables we see that, (i),(iii),(iv) have truth value T.

Note: Some basic Propositional Logic symbol, which was used in this question shown below-
The symbol $ \wedge $ denotes and, and $ \vee $ denotes or.
The symbol $ \to $denotes implication. The symbol $ \leftrightarrow $ indicates if and only if.