Question
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If p’s truth value is T and q’s truth value is F, then which of the following have the truth value T ?
i. $p \vee q$
ii. $ \sim p \vee q$
iii. $p \vee ( \sim q)$
iv. $p \wedge ( \sim q)$
A. (i),(ii),(iii)
B. (i),(iii),(iv)
C. (i),(ii),(iv)
D. (ii),(iii),(iv)

Answer
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Hint: Here, we will solve the given problem by considering each statements truth value and verify which compound statement is T i.e., Tautology.

Complete step-by-step answer:
i. $p \vee q$
Given,
The truth value of p is T and the truth value of q is F.
Now, we need to find the truth value of $p \vee q$ i.e.., “or” connectivity of statements of p and q
As, we know in the “or” connectivity if either of the statements p and q is $T$ then the compound statement $p \vee q$ will also be T
\[ p \vee q \\
  T \vee F \\
  T \\ \]
Hence, the truth value of $p \vee q$ is T i.e.., Tautology.

ii. $ \sim p \vee q$
Given,
The truth value of p is T and the truth value of q is F.
Therefore the truth value of ~p is F
Now, we need to find the truth value of $ \sim p \vee q$ i.e.., “or” connectivity of statements of negation (p) and q
\[ \Rightarrow \sim p \vee q \\
   \Rightarrow F \vee F \\
   \Rightarrow F \\ \]
Hence, the truth value of $ \sim p \vee q$ is F i.e.., Contradiction.

iii. $p \vee ( \sim q)$
Given,
The truth value of p is T and the truth value of q is F.
Therefore the truth value of ~q is T.
Now, we need to find the truth value of $p \vee ( \sim q)$ i.e.., “or” connectivity of statements of p and negation (q).
As, we know in the “or” connectivity if either of the statements is T then the compound statement will also be T
\[ \Rightarrow p \vee \sim q \\
   \Rightarrow T \vee T \\
   \Rightarrow T \\ \]
Hence, the truth value of $p \vee ( \sim q)$ is T i.e.., Tautology.

iv. $p \wedge ( \sim q)$
Given,
The truth value of p is T and the truth value of q is F.
Therefore the truth value of ~q is T.
Now, we need to find the truth value of $p \wedge ( \sim q)$ i.e.., “and” connectivity of statements of p and negation (q).
As we know in the “and” connectivity if the truth value of each statement is T then only the compound statement’s truth value will be T.
\[\Rightarrow p \wedge \sim q \\
   \Rightarrow T \wedge T \\
   \Rightarrow T \\ \]
Hence, the truth value of $p \wedge ( \sim q)$ is T i.e.., Tautology.
Therefore, (i), (iii), (iv) statements have the truth value T i.e.., Tautology.
Hence, the correct option for the given question is ‘B’.

Note: In solving the problems on truth values of statements if the connectivity is “or” then the truth value of the compound statement will be true if either of the two statements is true and if the connectivity is “and” then both of the statements should be true for the compound statement to be true.