Question

If \(\mathrm{p}=\mathrm{He}\) is intelligent, \(\mathrm{q}=\mathrm{He}\) is strong. Then symbolic form of the statement "It is wrong that he is intelligent or strong" is (a) \(\sim \mathrm{p} \mathrm{V} \sim \mathrm{q}\) (b) \(\sim(\mathrm{p} \Lambda \mathrm{q})\) (c) \(\sim(p \mathrm{~V} q)\) (d) \(\mathrm{p} \Lambda \sim \mathrm{q}\)

Short answer

The correct symbolic form of the given statement is: \( (c) \sim(p \mathrm{~V} q) \)

Step-by-step solution

Identify the connectives

First, let's identify which connectives we should use. In this case, the "OR" connective "∨" and the "NOT" connective "¬" are the relevant ones.

Translate the given statement into the symbolic form

Now, let's represent the given statement in a symbolic form. "It is wrong that he is intelligent or strong." We know that: p = He is intelligent q = He is strong The statement says that it is wrong that he is either intelligent or strong, so we will use the OR connective and the NOT connective: ¬(p ∨ q) Which means "NOT (He is intelligent OR He is strong)".

Compare the symbolic form with the given options

Let's compare the obtained symbolic form ¬(p ∨ q) with the given options: (a) ¬p ∨ ¬q (b) ¬(p ∧ q) (c) ¬(p ∨ q) (d) p ∧ ¬q Our symbolic form ¬(p ∨ q) matches option (c).

Final Answer

The correct symbolic form of the given statement is: (c) ¬(p ∨ q)

Key concepts

Logical Connectives

In the realm of symbolic logic, logical connectives are the backbone of building complex statements from simpler ones. They are akin to the conjunctions in a sentence, joining words and phrases to convey a more nuanced meaning.

There are several types of logical connectives, each with a unique role:

• The AND connective (denoted by \(\Lambda\)) joins two statements in a way that the resulting compound statement is true only if both individual statements are true.
• The OR connective (denoted by \(\vee\)) is inclusive, meaning the compound statement is true if at least one of the individual statements is true.
• The NOT connective (denoted by \(eg\)) negates the truth value of a single statement.

Understanding how to correctly use these connectives is essential in translating complex statements into their symbolic form, which can be a crucial skill not only in mathematics but also in fields like computer science and philosophy.

Negation

The concept of negation in symbolic logic is straightforward yet powerful. It is the operation that reverses the truth value of a proposition. Represented by the symbol \(eg\), negation transforms a true statement into a false one, and vice versa.

For instance, if we have a proposition \(p\) that stands for 'The sky is blue,' the negation of \(p\), written as \(eg p\), would mean 'The sky is not blue.' It’s essential to note that negation affects the entire statement it precedes. Hence, the expression \(eg(p \vee q)\) negates the entire compound statement formed by \(p\) and \(q\) connected by \('OR'\vee\), rather than negating each individually. This distinction is crucial in logical analysis and in understanding logical equivalences and contradictions.

Disjunction

The term disjunction refers to a compound statement formed using the logical OR connective, symbolized by \(\vee\). This connective creates a statement that is true if at least one of its component statements is true. For a disjunction involving two propositions \(p\) and \(q\), written as \(p \vee q\), the whole is true if either 'p is true, q is true, or both are true.

Disjunction is viewed as an inclusive 'OR,' distinguishing it from the exclusive 'OR,' which is true when exactly one of the statements is true, but not both. In mathematics and logic problems, unless stated otherwise, the 'OR' is typically inclusive. This knowledge helps in accurately breaking down complex statements and can play a crucial role in various reasoning tasks, such as determining the validity of arguments or solving logical puzzles.