Question

Which of the following is not true?
A. If the last column of its truth table contains only F then it is a contradiction.
B. Negation of a negation of a statement is the statement itself.
C. If p and q are two statements then \[p\Leftrightarrow q\] is a tautology.
D. If the last column of its truth table contains only T then it is a tautology.

Answer 1

Hint: In the given question, we are asked which of the following options given to us. We will go through each of the options given to us and check the logic of the statement whether it is correct or not. Then we will get the statement which is incorrect. Hence, we will have the statement which is not true.

Complete step by step solution:
According to the given question, we are asked to find the incorrect statement from the given options.
We will be going through each of the given options and check if the statements are correct or not.
First, we have the statement, ‘If the last column of its truth table contains only F then it is a contradiction’. This particular statement is correct as the truth table has all the possibilities that T and F can have. It starts with T in the first row and in the last row F and F gives T. If in the last row all are F, then it is a contradiction. So, it is correct.
Next, we have, ‘Negation of a negation of a statement is the statement itself’. This statement is correct and makes sense too. If we write the negation of a statement and further we write the negation of the previously written negative statement, we will get the initial statement. So, this statement is correct.
Next, ‘If p and q are two statements then \[p\Leftrightarrow q\] is a tautology’. This statement is incorrect as tautology refers to the statements that remain true in all circumstances. For example – Desert is a dry place. But in a truth table the last row has both T and F, so it is not a tautology. So, it is an incorrect option.

\[p\]	\[q\]	\[p\Leftrightarrow q\]
T	T	T
T	F	F
F	T	F
F	F	T

Next, we have, ‘If the last column of its truth table contains only T then it is a tautology’. This statement is true as from the above table, the last row of a truth table has one T. So, it is correct.
Therefore, option C. If p and q are two statements then \[p\Leftrightarrow q\] is a tautology.

Note: While dealing with truth tables, have a decent idea about how it appears, else there would not seem any mistakes in any of the statements. The tautology based question has one of the parameters given using which we have to confirm whether the statement is correct or not.