Question

# AIf p: 3 is a prime number and q: one plus one is three, then the compound statement "It is not that 3 is a prime number or it is not that one plus one is three" is:\begin{align} & \left( a \right)\neg p\vee q \\ & \left( b \right)\neg \left( p\vee q \right) \\ & \left( c \right)p\wedge \neg q \\ & \left( d \right)\neg p\vee \neg q \\ & \left( e \right)p\vee \neg q \\ \end{align}

Hint: In this question, we need to look at the Boolean algebra to solve it. First look at the basic definitions and then try to write the elementary operation of logic for the given statement.

Let us look at some of the basic definitions of Boolean algebra.
LOGIC: Logic is the subject that deals with the method of reasoning. It provides us rules for determining the validity of a given argument in proving a theorem.
STATEMENT: A statement is an assertive sentence which is either true or false but not both a true statement is called a valid statement. Otherwise it is called invalid statement. Statements are denoted by small letters i.e. p, q, r etc.
Example: p: A triangle has four sides.
SIMPLE STATEMENT: A statement which cannot be broken into two or more statements is called a simple statement.
OPEN STATEMENT: A sentence which contains one or more variables such that when certain values are given to the variable it becomes a statement, is called an open statement.
COMPOUND STATEMENT: If two or more simple statements are combined by the use of words such as 'and', 'or', 'not', 'if', 'then', 'if and only if', then the resulting statement is called a compound statement.
TRUTH VALUE: A statement can be either 'true' or 'false' which is called truth value of a statement.
ELEMENTARY OPERATIONS OF LOGIC
NEGOTIATION: A statement which is formed by changing the truth value of a given statement by using the word like 'no', 'not' is called negotiation of a given statement. If p is a statement, then negotiation is denoted by $\neg p$ .
CONJUNCTION: A compound sentence formed by two simple sentences p and q using connective 'and' is called the conjunction of p and q and it is represented by $p\wedge q$.
DISJUNCTION: A compound sentence formed by two simple sentences p and q using connective 'or' is called the conjunction of p and q and it is represented by $p\vee q$.
p: 3 is a prime number and q: one plus one is three
Given the statement "It is not that 3 is a prime number or it is not that one plus one is three".
The first half of the sentence is the negotiation of statement p and the second half of the sentence is the negotiation of statement q.
As the term 'or' is used in the final statement it means that it means that the compound statement is the disjunction of the statements negotiation of p and negotiation of q.
Thus, it is represented as:
$\neg p\vee \neg q$
Hence, the correct option is (d).

Note: The word 'not' present in the compound statement plays an important role because if we do not consider that word not then the compound statement will become just a disjunction of statements p and q which will be represented as $p\vee q$. Similarly, if we consider 'and' in the place of 'or' then the compound statement changes as the conjunction of the statements negotiation of p and negotiation of q which is represented as: $\neg p\wedge \neg q$ The word 'not' present in the compound statement twice indicates the negotiation for both the statements p and q neglecting any one of them gives us the incorrect option.