Question

Negation of \(\mathrm{p} \rightarrow \mathrm{q}\) is (a) \(\sim \mathrm{p} \rightarrow \sim \mathrm{q}\) (b) \(\mathrm{p} \rightarrow \sim \mathrm{q}\) (c) \(\mathrm{p} \rightarrow \mathrm{q}\) (d) \(\sim \mathrm{q} \rightarrow \sim \mathrm{p}\)

Short answer

The closest answer to the negation of \(\mathrm{p} \rightarrow \mathrm{q}\) is (b) \(\mathrm{p} \rightarrow \sim \mathrm{q}\), although it's not equivalent to the negated expression \((\mathrm{p} \wedge \sim \mathrm{q})\).

Step-by-step solution

Understand the Logical Connectives

In propositional logic, logical connectives are used to link different propositions/statement variables. Here we have two connectives: 1. Implication (denoted by \(\rightarrow\)): the implication \(\mathrm{p} \rightarrow \mathrm{q}\) means 'if p, then q' or 'p implies q'. This connective is true unless p is true and q is false. 2. Negation (denoted by \(\sim\)): the negation \(\sim \mathrm{p}\) means 'not p'. This connective simply reverses the truth value of the proposition it's applied to.

Rewrite the Implication using Disjunction and Negation

Recall that an implication can also be written as a disjunction (logical OR, denoted by \(\vee\)) along with negation, as follows: \(\mathrm{p} \rightarrow \mathrm{q} \equiv \sim \mathrm{p} \vee \mathrm{q}\) This means that the implication \(\mathrm{p} \rightarrow \mathrm{q}\) is logically equivalent to the statement 'either p is false or q is true'.

Negate the Equivalent Expression

Now, we will negate the equivalent expression of the implication \(\sim \mathrm{p} \vee \mathrm{q}\), using the De Morgan's law, which states that: \(\sim (\mathrm{A} \vee \mathrm{B}) \equiv (\sim \mathrm{A} \wedge \sim \mathrm{B})\) \(\sim (\sim \mathrm{p} \vee \mathrm{q}) \equiv (\sim (\sim \mathrm{p}) \wedge \sim \mathrm{q})\) Applying double negation (as \(\sim(\sim \mathrm{A})\) is the same as \(\mathrm{A}\)): \(\sim (\sim \mathrm{p} \vee \mathrm{q}) \equiv (\mathrm{p} \wedge \sim \mathrm{q})\)

Identify the Correct Option

Finally, let's compare the negated expression \((\mathrm{p} \wedge \sim \mathrm{q})\) with the provided options: (a) \(\sim \mathrm{p} \rightarrow \sim \mathrm{q}\) (b) \(\mathrm{p} \rightarrow \sim \mathrm{q}\) (c) \(\mathrm{p} \rightarrow \mathrm{q}\) (d) \(\sim \mathrm{q} \rightarrow \sim \mathrm{p}\) None of these options exactly match our negated expression \((\mathrm{p} \wedge \sim \mathrm{q})\). However, (b) is the closest because it also states that \(\mathrm{p}\) is true and \(\mathrm{q}\) is false, though using implication notation. It's important to note that (b) isn't equivalent to the negated expression but is the closest option provided. Thus, we can choose the closest answer, which is - (b) \(\mathrm{p} \rightarrow \sim \mathrm{q}\).

Key concepts

Logical Connectives

In propositional logic, logical connectives are essential tools that help us form compound propositions from simpler ones. These connectives include symbols like \( \land \) (and), \( \lor \) (or), \( \sim \) (not), and \( \rightarrow \) (implies). By linking propositions together, we can express more complex logical ideas.

• And (\( \land \)): This is true only if both propositions are true.
• Or (\( \lor \)): This is true if at least one of the propositions is true.
• Not (\( \sim \)): This changes the truth value of a proposition.
• Implies (\( \rightarrow \)): This is true unless a true proposition implies a false one.

Understanding these connectives enables us to manipulate and evaluate logical statements effectively.

Implication

The implication is a logical connective that takes the form \( \mathrm{p} \rightarrow \mathrm{q} \), read as "if \( \mathrm{p} \) then \( \mathrm{q} \)." This statement is only false if \( \mathrm{p} \) is true while \( \mathrm{q} \) is false. Otherwise, the implication is true. Another way to think about implication is to assume that if the first part (\( \mathrm{p} \), the antecedent) is true, it must lead to the second part (\( \mathrm{q} \), the consequent).

• Example: "If it rains, then the ground will be wet." This is only false if it rains, and the ground is not wet.

Additionally, through logical equivalence, we can express the implication as a disjunction: \( \mathrm{p} \rightarrow \mathrm{q} \equiv \sim \mathrm{p} \vee \mathrm{q} \), which simplifies analysis in logic.

Negation

Negation in logic is simple but powerful. By applying the negation operator \( \sim \), the truth value of a proposition is reversed. This means:

• If something is true, its negation is false.
• If something is false, its negation is true.

For example, if \( \mathrm{p} \) represents the statement "The sky is blue," then \( \sim \mathrm{p} \) would stand for "The sky is not blue."
Negation is fundamental when working with more complex logical expressions, as it allows for the transformation of statements, such as using De Morgan's Laws to negate compound propositions.

De Morgan's Law

De Morgan's Laws are critical rules that help us understand the negation of complex propositions involving AND and OR. These laws look like this:

• The negation of a disjunction: \( \sim(\mathrm{A} \vee \mathrm{B}) \equiv (\sim \mathrm{A} \wedge \sim \mathrm{B}) \)
• The negation of a conjunction: \( \sim(\mathrm{A} \wedge \mathrm{B}) \equiv (\sim \mathrm{A} \vee \sim \mathrm{B}) \)

These transformations show how negating a compound expression affects its internal components.
For example, given a disjunction \( \mathrm{p} \vee \mathrm{q} \), it can be negated to \( \sim \mathrm{p} \wedge \sim \mathrm{q} \). This is instrumental in defining logical equivalences, simplifying expressions, and proving logical theorems.Using these laws greatly aids in converting or understanding expressions in logic, such as in the original exercise where \( \sim(\sim \mathrm{p} \vee \mathrm{q}) \equiv (\mathrm{p} \wedge \sim \mathrm{q}) \).