Question

The negation of the compound proposition \[p\vee \left( \sim p\vee q \right)\] is:
\[\left( \text{a} \right)\text{ }p\wedge q\]
(b) t
(c) f
\[\left( \text{d} \right)\text{ }\left( p\wedge q \right)\wedge \sim p\]

Answer 1

Hint: To solve the question given above, we will first write the truth tables of p and truth tables of q. Then with the help of the truth table of p, we will find the truth table of \[\sim p.\] Then with the help of these truth tables, we will find the truth table of the term given in the question. Then we will find the negation of this truth table.

Complete step by step solution:
We will take the help of the truth tables to solve this question. A truth table is a mathematical table used to determine if a compound statement is true or false. In a truth table, each element is typically represented by a letter or a variable and each statement has its corresponding column in the truth table that lists all of the possible truth values. Now, we will write the truth table of p:

\[p\]

T

T

F

F

Similarly, the truth table of q.

\[q\]

T

F

T

F

Now, the negation of p that is, \[\sim p\] is given by:

\[\sim p\]

F

F

T

T

Now, we will find the value of \[\sim p\vee q.\] This term means that \[\sim p\] or q. This term will be true if either \[\sim p\] is true or q is true or both are true. It is false only if both \[\sim p\] and q are false. Thus, we will get,

\[\sim p\]	\[q\]	\[\sim p\vee q\]
F	T	T
F	F	F
T	T	T
T	F	T

Now, we will find the value of \[p\vee \left( \sim p\vee q \right).\] This term means that p or \[\left( \sim p\vee q \right)\] is true or both are true. It is false only if both p and \[\left( \sim p\vee q \right)\] are false. Thus, we will get,

\[p\]	\[\sim p\vee q\]	\[p\vee \left( \sim p\vee q \right)\]
T	T	T
T	F	T
F	T	T
F	T	T

Now we know that the truth table is true for every statement, so it will be a tautology. Now, the negation of tautology is a fallacy. So, the negation of the term is a fallacy.
Hence, the option (c) is the right answer.

Note: The above question can also be solved in an alternate method. We have to find the negation of \[p\vee \left( \sim p\vee q \right)\] i.e. \[\sim \left[ p\vee \left( \sim p\vee q \right) \right].\] We can also write this as
\[\Rightarrow \sim \left[ \left( p\vee \sim p \right)\vee \left( p\vee q \right) \right]\]
\[\Rightarrow \sim \left[ t\vee \left( p\vee q \right) \right]\]
\[\Rightarrow \sim \left( t \right)\]
\[=f\]